CoCalc Public Fileswww / shwayder / paper / 91rpaper.texOpen in CoCalc with one click!
Author: William A. Stein
1
\documentclass[12pt]{article}
2
\usepackage{latexsym}
3
\usepackage{amsmath}
4
\usepackage{amsfonts}
5
\usepackage{amssymb}
6
\usepackage{fancyhdr}
7
\usepackage{graphicx}
8
\usepackage{pstricks}
9
\pagestyle{fancy}
10
\fancyhf{}
11
\lhead{Shwayder}
12
\rhead{\thepage}
13
14
\newif\ifpdf
15
\ifx\pdfoutput\undefined
16
\pdffalse % we are not running PDFLaTeX
17
\else
18
\pdfoutput=1 % we are running PDFLaTeX
19
\pdftrue
20
\fi
21
22
\ifpdf
23
\usepackage[pdftex]{graphicx}
24
\else
25
\usepackage{graphicx}
26
\fi
27
28
\DeclareMathOperator{\Z}{\mathbb{Z}}
29
\DeclareMathOperator{\Q}{\mathbb{Q}}
30
\DeclareMathOperator{\R}{\mathbb{R}}
31
\DeclareMathOperator{\C}{\mathbb{C}}
32
\newcommand{\Zmod}[1]{\mathbb{Z}/#1\mathbb{Z}}
33
34
\newtheorem{theorem}{Theorem}
35
\newtheorem{corollary}[theorem]{Corollary}
36
\newtheorem{definition}{Definition}
37
38
\title{Visualizing $L(E,s)$}
39
\author{Ariel Shwayder}
40
\date{May 14, 2002}
41
\begin{document}
42
43
\ifpdf
44
\DeclareGraphicsExtensions{.pdf, .jpg, .tif}
45
\else
46
\DeclareGraphicsExtensions{.eps, .jpg}
47
\fi
48
\maketitle
49
%Opening Graph
50
\begin{center}
51
\psset{unit=.75in}
52
\begin{pspicture}(-0.5,-1.5)(3,1.5)
53
\psgrid[gridcolor=gray]
54
55
% axes
56
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
57
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
58
59
\psline[linecolor=blue]
60
(0.01999999999999999999999999999,0.1282118407846508175583106774)
61
(0.03999999999999999999999999999,0.2373160924769400586971505305)
62
(0.05999999999999999999999999999,0.3290692646267834227947960365)
63
(0.07999999999999999999999999999,0.4051228825645450585304885008)
64
(0.09999999999999999999999999999,0.4670247962483374987539977831)
65
(0.1199999999999999999999999999,0.5162212238980464124901938755)
66
(0.1399999999999999999999999999,0.5540593933614910786360224599)
67
(0.1599999999999999999999999999,0.5817906609306697377006666278)
68
(0.1799999999999999999999999999,0.6005740024782968079515816309)
69
(0.1999999999999999999999999999,0.6114797854386769375466728906)
70
(0.2199999999999999999999999999,0.6154937424288827048452444026)
71
(0.2399999999999999999999999999,0.6135210783070263808904877223)
72
(0.2599999999999999999999999999,0.6063906522990695460707949904)
73
(0.2799999999999999999999999999,0.5948591855930355805098153124)
74
(0.2999999999999999999999999999,0.5796154525926131415502242463)
75
(0.3199999999999999999999999999,0.5612844209279497530891884273)
76
(0.3399999999999999999999999999,0.5404313114210843311436354732)
77
(0.3599999999999999999999999999,0.5175655545724288943300248607)
78
(0.3799999999999999999999999999,0.4931446248429705007442634947)
79
(0.3999999999999999999999999999,0.4675777381191454062641922725)
80
(0.4199999999999999999999999999,0.4412294013233299353321305827)
81
(0.4399999999999999999999999999,0.4144228062275087451541479708)
82
(0.4599999999999999999999999999,0.3874430621913117241632443584)
83
(0.4799999999999999999999999999,0.3605402648243859340416994045)
84
(0.5000000000000000000000000000,0.3339323995091193670233369006)
85
(0.5199999999999999999999999999,0.3078080803514474329469164284)
86
(0.5399999999999999999999999999,0.2823291264897498143293603855)
87
(0.5599999999999999999999999999,0.2576329788163385994174375024)
88
(0.5799999999999999999999999999,0.2338349610813938431173462811)
89
(0.5999999999999999999999999999,0.2110303900812833649030031164)
90
(0.6199999999999999999999999999,0.1892965402053072762858670989)
91
(0.6400000000000000000000000000,0.1686944680479713779357549229)
92
(0.6600000000000000000000000000,0.1492707031066874146744069022)
93
(0.6800000000000000000000000000,0.1310588107941084029506542038)
94
(0.6999999999999999999999999999,0.1140808341151073404642970743)
95
(0.7200000000000000000000000000,0.09834862040401863699141623165)
96
(0.7400000000000000000000000000,0.08386503950000162025807057460)
97
(0.7600000000000000000000000000,0.07062509966770839602136767703)
98
(0.7800000000000000000000000000,0.05861696745606273048447558611)
99
(0.8000000000000000000000000000,0.04782289753798470487571183686)
100
(0.8200000000000000000000000000,0.03822007839542337087786988235)
101
(0.8400000000000000000000000000,0.02978139951327671835061570502)
102
(0.8600000000000000000000000000,0.02247614552806240305228065712)
103
(0.8800000000000000000000000000,0.01627062254720350685508406602)
104
(0.9000000000000000000000000000,0.01112872161651056641114606877)
105
(0.9200000000000000000000000000,0.007012424070294267547088399575)
106
(0.9400000000000000000000000000,0.003882253253437653182664720498)
107
(0.9600000000000000000000000000,0.001697676860141058292114259975)
108
(0.9800000000000000000000000000,0.0004174638919724419127585133029)
109
(1.000000000000000000000000000,0)
110
(1.020000000000000000000000000,0.0004035647435303338322320539744)
111
(1.040000000000000000000000000,0.001586574072428146472169781739)
112
(1.060000000000000000000000000,0.003507791122018288156933516286)
113
(1.080000000000000000000000000,0.006126508199826910685548956201)
114
(1.100000000000000000000000000,0.009402702642385630228987141858)
115
(1.120000000000000000000000000,0.01329716902832726856943509757)
116
(1.140000000000000000000000000,0.01777163005123880089972840215)
117
(1.160000000000000000000000000,0.02278882818228977135119888984)
118
(1.180000000000000000000000000,0.02831260008851170488116623996)
119
(1.200000000000000000000000000,0.03430793561767636946629859578)
120
(1.220000000000000000000000000,0.04074102301485355415349698977)
121
(1.240000000000000000000000000,0.04757928189871473307592052701)
122
(1.260000000000000000000000000,0.05479138539723744107672509691)
123
(1.280000000000000000000000000,0.06234727272237282934341842687)
124
(1.300000000000000000000000000,0.07021815335115663505273155096)
125
(1.320000000000000000000000000,0.07837650387634396617326729041)
126
(1.340000000000000000000000000,0.08679605849259037625151631419)
127
(1.360000000000000000000000000,0.09545179399413744425048307793)
128
(1.380000000000000000000000000,0.1043199100765387671527013774)
129
(1.400000000000000000000000000,0.1133778056578303296415459023)
130
(1.420000000000000000000000000,0.1226040518633591809553073792)
131
(1.440000000000000000000000000,0.1319783622528933898669772174)
132
(1.460000000000000000000000000,0.1414815608083091813618921277)
133
(1.480000000000000000000000000,0.1510955481447620810974098848)
134
(1.500000000000000000000000000,0.1608032663574824684779060275)
135
(1.520000000000000000000000000,0.1705886628698883736845417274)
136
(1.540000000000000000000000000,0.1804366536062881212790460355)
137
(1.560000000000000000000000000,0.1903330857737737579051318979)
138
(1.580000000000000000000000000,0.2002647005027174057571610452)
139
(1.600000000000000000000000000,0.2102190955633242676617659950)
140
(1.620000000000000000000000000,0.2201846883467286991995015459)
141
(1.640000000000000000000000000,0.2301506792729173891861817815)
142
(1.660000000000000000000000000,0.2401070157641129847479222917)
143
(1.680000000000000000000000000,0.2500443569009518146360725463)
144
(1.700000000000000000000000000,0.2599540388596523353920034575)
145
(1.720000000000000000000000000,0.2698280412112200659832513937)
146
(1.740000000000000000000000000,0.2796589541484050598926959187)
147
(1.760000000000000000000000000,0.2894399466924653601976240572)
148
(1.780000000000000000000000000,0.2991647359196508859241351051)
149
(1.800000000000000000000000000,0.3088275572365733337781196326)
150
(1.820000000000000000000000000,0.3184231357241450149813815388)
151
(1.840000000000000000000000000,0.3279466585614381944502036829)
152
(1.860000000000000000000000000,0.3373937485335301255881053191)
153
(1.880000000000000000000000000,0.3467604386210593163719238026)
154
(1.900000000000000000000000000,0.3560431476637349571773712703)
155
(1.920000000000000000000000000,0.3652386570853303616433253836)
156
(1.940000000000000000000000000,0.3743440886636758844003824977)
157
(1.960000000000000000000000000,0.3833568833257765112024623199)
158
(1.980000000000000000000000000,0.3922747809453494444043683176)
159
(2.000000000000000000000000000,0.4010958011177482601091397234)
160
(2.020000000000000000000000000,0.4098182248853583997139466982)
161
(2.040000000000000000000000000,0.4184405773850644032129103672)
162
(2.060000000000000000000000000,0.4269616113882572976601743283)
163
(2.080000000000000000000000000,0.4353802917030298809849036790)
164
(2.100000000000000000000000000,0.4436957804076610005384183947)
165
(2.120000000000000000000000000,0.4519074228841834951845159420)
166
(2.140000000000000000000000000,0.4600147346207336244999482509)
167
(2.160000000000000000000000000,0.4680173887514648687732405402)
168
(2.180000000000000000000000000,0.4759152043030509777621529758)
169
(2.200000000000000000000000000,0.4837081351171795913522571217)
170
(2.220000000000000000000000000,0.4913962594189284500945307527)
171
(2.240000000000000000000000000,0.4989797700015030469327185371)
172
(2.260000000000000000000000000,0.5064589649984813437454309018)
173
(2.280000000000000000000000000,0.5138342392154434340222207852)
174
(2.300000000000000000000000000,0.5211060759936489147101614380)
175
(2.320000000000000000000000000,0.5282750395792508246078778271)
176
(2.340000000000000000000000000,0.5353417679723922206432863952)
177
(2.360000000000000000000000000,0.5423069662314109008644204260)
178
(2.380000000000000000000000000,0.5491714002082716328220645855)
179
(2.400000000000000000000000000,0.5559358906922466788124893554)
180
(2.420000000000000000000000000,0.5626013079397684826686378747)
181
(2.440000000000000000000000000,0.5691685665692779537194967325)
182
(2.460000000000000000000000000,0.5756386208007834293494110199)
183
(2.480000000000000000000000000,0.5820124600207253421507697039)
184
(2.500000000000000000000000000,0.5882911046536066645927086058)
185
(2.520000000000000000000000000,0.5944756023226966756297476987)
186
(2.540000000000000000000000000,0.6005670242829432747398377343)
187
(2.560000000000000000000000000,0.6065664621100351564554136207)
188
(2.580000000000000000000000000,0.6124750246303382151808133756)
189
(2.600000000000000000000000000,0.6182938350771894623083546158)
190
(2.620000000000000000000000000,0.6240240284597656772586630088)
191
(2.640000000000000000000000000,0.6296667491314524040321696123)
192
(2.660000000000000000000000000,0.6352231485453213879490153223)
193
(2.680000000000000000000000000,0.6406943831849809578932403951)
194
(2.700000000000000000000000000,0.6460816126596941982135622091)
195
(2.720000000000000000000000000,0.6513859979532641654857877552)
196
(2.740000000000000000000000000,0.6566086998167641554676401766)
197
(2.760000000000000000000000000,0.6617508772957444860683841975)
198
(2.780000000000000000000000000,0.6668136863830758922830232224)
199
(2.800000000000000000000000000,0.6717982787890939613622618111)
200
(2.820000000000000000000000000,0.6767058008211896637742053670)
201
(2.840000000000000000000000000,0.6815373923654485991297106573)
202
(2.860000000000000000000000000,0.6862941859633767558735804127)
203
(2.880000000000000000000000000,0.6909773059771640881089063440)
204
(2.900000000000000000000000000,0.6955878678373297725922876254)
205
(2.920000000000000000000000000,0.7001269773669653681322587831)
206
(2.940000000000000000000000000,0.7045957301771450108972006522)
207
(2.960000000000000000000000000,0.7089952111284059978729044088)
208
(2.980000000000000000000000000,0.7133264938535193905197729615)
209
(3.000000000000000000000000000,0.7175906403370693594735075744)
210
\rput(3.7,0.75){$[0,-1,1,0,2]$}
211
\pscircle[linecolor=red](1,0){0.1}
212
\end{pspicture}
213
214
\end{center}
215
216
\tableofcontents
217
\pagebreak
218
\section{Why bother to do this?}
219
220
Elliptic curves, and the machinery involved in them have been a hot topic in modern mathematics for quite some time. They came to the fore in the public consciousness most prevalently because of their deep involvement in Andrew Wiles's 1994 proof of Fermat's Last Theorem. This is in fact when I first became aware of them. Elliptic curves have become popular perhaps not only because of their deep and interesting properties, but also because of their fairly simple definition. The notion of an $L$-series attached to an elliptic curve is also a fairly simple notion (as will be explained below), but the study of the correlation between these two objects has led to many of the biggest unsolved problems in mathematics today.
221
222
The Birch and Swinnerton-Dyer (BSD) conjecture is the statement that the rank (a simple algebraic invariant) of an elliptic curve is equal to the order of vanishing at zero (a simple analytic property) of the $L$-series attached to that curve. The conjecture was first formulated in the early 1960s, and today we still don't have a good way of approaching the problem. With Fermat's Last Theorem, even before it was proved, the conjecture was known to hold for many specific cases. With BSD we don't even know how to show it is true for some very seemingly simple cases (for example, curves of rank 4).
223
224
Until 1986 it was not even known if the BSD conjecture held for a curve as simple as $y^2+y=x^3-7x+6$. In fact, the proof that it did hold was so deep that it provided an effective solution to the Gauss class number problem. A seemingly simple problem such as showing that there existed an elliptic curve whose $L$-series had order 3 was a very deep theorem of Gross and Zagier, and as of this writing, it is an open problem to prove that there is an elliptic curve whose $L$-series has order 4 or higher.
225
226
It is with these questions in mind that we approach the idea of graphing the $L$-series attached to elliptic curves. While we doubt that anything can be proven through pictures, having pictures as a reference is a very helpful tool when dealing with these series, and perhaps these pictures will give us more insight into what is happening with $L(E,s)$.
227
228
\section{A quick introduction to Elliptic Curves and the series associated with them}
229
\subsection{All about $E$}
230
The definition of an elliptic curve is the following: An elliptic curve~$E$ is a cubic curve of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where the $a_i$'s are constants from a field $K$. We define the discriminant of the curve as
231
$$\Delta = -b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6$$ with $$b_2=a_1^2+4a_2, \quad b_4 = 2a_4+a_1a_3, \quad b_6=a_3^2+4a_6.$$ We require that $\Delta\neq 0$. For ease of notation we write $[a_1,a_3,a_2,a_4,a_6]$ when referring to the curve $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$.
232
233
Perhaps one of the most interesting properties of an elliptic curve is that the points on an elliptic curve form a group. The key step in seeing this is to note that given any two points on an elliptic curve we can, in a natural way, define what it means to ``add" those two points together. On an intuitive level, when you add two points together you draw a line connecting them, and see where that line intersects the elliptic curve. If there is no third point of intersection then we say that those two points add to ``the point at infinity," which is the identity element in the group. Otherwise, if they intersect at a third point, $(x,y)$, then we define that the ``addition" of those two points to be $(x,-y)$. (The reason that you need to switch the $y$-coordinate is so that all of the group axioms come out correctly.)
234
235
Once you determine that the points do indeed form a group, then the natural question to ask is, what is this group? If we are considering our curve over $\R$ then the story becomes less interesting. In this case the group is infinite, as given any $x\in\R$ one can find $y\in\R$ such that $(x,y)$ lies on the curve. It being infinite is not what makes it uninteresting, but it is the fact that the group is infinitely generated that makes it so. Because it is infinitely generated there is not much we can say about the group. In fact for any elliptic curve, the group $E(\R)$ is always either $S^1$ or $S^1\times\Zmod{2}$. Where $S^1$ is the group formed by the points in the complex plane on the circle of radius 1 under multiplication.
236
237
Since the problem of $E(\R)$ has been solved, we turn our attention to what happens when we consider our curve over $\Q$. In this case Mordell's theorem tells us that the group $E(\Q)$ is finitely generated and hence that $$E(\Q) \simeq \Z^r \times E(\Q)_{\rm tors}$$ where $r$ is a non-negative integer, and $E(\Q)_{\rm tors}$ is the finite subgroup of points of finite order in $E(\Q)$. Professor Barry Mazur, in a 1976 theorem, showed that $E(\Q)_{\rm tors}$ must be isomorphic to one of the following groups:
238
\begin{eqnarray*}
239
\Z/n\Z, && {\rm for}\ n \leq 10\ {\rm or}\ n = 12.\\
240
(\Z/2\Z) \times (\Z/2n\Z), && {\rm for}\ n \leq 4.
241
\end{eqnarray*}
242
243
The integer $r$ is called the rank of the elliptic curve.
244
It is a folklore conjecture that $r$ can be arbitrarily large, however, the current record is a curve of rank at least 24. This was discovered by Roland Martin and William McMillen of the National Security Agency in January 2000.\\ (See {\footnotesize\tt http://listserv.nodak.edu/scripts/wa.exe?A2=ind0005\&L=nmbrthry\&P=R182})
245
246
\subsection{The mysterious $L$}
247
To every elliptic curve one can attach a certain series that we call $L(E,s)$. To define $L(E,s)$ recall that we have previously discussed points on an elliptic curve over such fields as $\Q$ or $\R$. However, the notion of points on an elliptic curve is not limited to these fields. One can consider the number of points on an elliptic curve over $\Zmod{p}$ for any prime $p$ (that does not divide the discriminant of the curve). We denote the number of points on $E$ over $\Zmod{p}$ as $\#E(\Zmod{p})$. We can now define a sequence of numbers $a_p$ such that $a_p = p+1-\#E(\Zmod{p})$. There is also a slightly more complicated way to define $a_n$ for any number. These $a_n$ can be found in PARI by using the {\tt ellan} command.
248
249
Once we have these $a_n$ we can now define $L(E,s)$:
250
$$L(E,s) = \sum_{n=1}^\infty a_n n^{-s}$$
251
252
It is a theorem of Breuil, Conrad, Diamond, Taylor, and Wiles that $L(E,s)$ can be extended to an analytic function on all of $\C$. As with any other analytic function we can ask what the order of vanishing of $L(E,s)$ is at any point. It turns out that the order of vanishing of $L(E,s)$ at $s=1$ is a rather interesting story. In fact the Birch and Swinnerton-Dyer conjecture is that the order of vanishing at $s=1$ is exactly equal to the rank of the elliptic curve.
253
254
In other words, for any elliptic curve, $E$, $$L(E,s) = k(s-1)^r + \textrm{higher order terms}$$ at $s=1$. Where here $k \neq 0$ and $r$ is such that $E(\Q) \simeq \Z^r \times E(\Q)_{\rm tors}$.
255
256
The BSD conjecture is fairly amazing in that it asserts the equality of two seemingly very different quantities.
257
258
So far, the BSD conjecture has been proved when ord$_{s=1}L(E,s)\leq 1$ by Gross, Kolyvagin, Zagier, et al. However, for ord $>1$ it is still an open problem, and as was mentioned above, it has yet to be proven that any elliptic curve has rank 4.
259
260
261
\subsection{How I learned to stop worrying and love the $\Lambda$}
262
263
Using our $L$-series, and in fact using any $L$-series one can define the notion of a $\Lambda$ function that is very similar to $L(E,s)$, except that it has more symmetries. It is defined as follows: $$\Lambda(Es) = N^{\frac{s}{2}}(2\pi)^{-s}\Gamma(s) L(E,s).$$ Where $N$ is the conductor of the curve and $\Gamma(s)$ is the complete $\Gamma$ function evaluated at $s$.
264
265
However, when only considering those $L$-series that come from elliptic curves the associated $\Lambda$-series obeys the following symmetry:$$\Lambda(E,s) = \varepsilon\Lambda(E, 2-s)$$ where $\varepsilon \in \{\pm 1\}$ is the root number of $E$ (which can be found using {\tt ellrootno} in PARI). Because of this symmetry, the graphs of $\Lambda(E,s)$ can look ``nicer'' then those of $L(E,s)$ and in the graphs below we produce graphs of both $L(E,s)$ and $\Lambda(E,s)$ for this reason.
266
267
\section{The formulas and methods used for these graphs}
268
The question that then needs to be asked is ``What does $L(E,s)$ look like?" This is the question that we set about to answer. In order to do so we use the free program PARI (available online at {\tt http://www.parigp-home.de/}) to generate a list of points which can then be graphed. Fortunately, PARI has a nice built-in feature for computing $L(E,s)$ which makes the process much easier.
269
270
The function used to output the $L$-series data was:
271
\begin{verbatim}
272
{printellseries(fname, curve) =
273
E = ellinit(curve);
274
for(x=1,150,
275
s=0.0+x/50;
276
write(fname,"(",s,",",
277
nice(elllseries(E,s,1)),")"));
278
}
279
\end{verbatim}
280
Where the function {\tt nice} is:
281
\begin{verbatim}
282
nice(x)=if(abs(x)<(10^(-25)), return(0), return(x))
283
\end{verbatim}
284
This {\tt nice} function is necessary because otherwise if the value at a certain point is too small, PARI will output in scientific notation, which makes the data unreadable by the program used to graph it. This way, values that are below a certain tolerance are simply converted to 0.
285
286
The {\tt printellseries} function takes as input a filename and a vector defining a curve. It then outputs a list of points for the $L$-series of that curve, computed at intervals of .02 to the file \texttt{fname}.
287
288
Using this function, along with a little ingenious shell scripting and the help of the pstricks package of \LaTeX, we were able to generate the graphs seen in section 4 along with many others.
289
290
This method worked well for graphing $L(E,s)$ for real values of $s$. However, $L(E,s)$ is in reality a complex analytic function, so it is defined for any complex value $s$ as well. To solve this problem we could not use the built-in {\tt elllseries} function since it was not able to compute $L(E,s)$ for complex-valued~$s$.
291
292
In order to compute $L(E,s)$, we look to the following formula:
293
$$ L(E,s) = N^{\frac{-s}{2}}\cdot (2\pi)^s\cdot \Gamma(s)^{-1} \cdot \sum_{n=1}^\infty a_n\cdot (F_n(s-1)-\varepsilon F_n(1-s)).$$
294
Here $N$ is the conductor of the curve, $\Gamma$ is the standard $\Gamma$-function, $\varepsilon$ is as above, and $$F_n(t) = \Gamma\left(t+1, \frac{2\pi n}{\sqrt{N}}\right)\cdot \left(\frac{\sqrt{N}}{2\pi n}\right)^{t+1}.$$ In the formula for $F_n(t)$, $\Gamma(x,y)$ is the standard incomplete $\Gamma$ function.
295
This formula comes from the solution exercise 24 on page 521 in chapter 10.4 of Henri Cohen's book \emph{Advanced Topics in Computational Number Theory} (Springer-Verlag, March 2000).
296
297
The main problem in using PARI for these computations was that the implementation of the $\Gamma$-function in PARI does not include complex-valued arguments. For a while we played with trying to use other formulas to represent $\Gamma$ so that PARI could be used. In the end, however, we discovered that the $\Gamma$-function implementation in Mathematica includes the ability to compute for complex-valued arguments, and so we decided to use Mathematica for that part of the computation and simply used the formulas as listed above. In addition, Mathematica has the ability to output three-dimensional graphs.
298
299
\section{Graphing $L, \Lambda: \R \rightarrow \R$}
300
Figure~\ref{rank11} is a graph of the $L$-series for the curve $y^2-y=x^3+x^2-10x-20$. This curve has conductor 11 which is the curve of smallest conductor.
301
\begin{figure}
302
%\vspace{0.2in}
303
304
\begin{center}
305
\psset{unit=.9in}
306
\begin{pspicture}(-0.5,-1.5)(3,1.5)
307
\psgrid[gridcolor=gray]
308
309
% axes
310
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
311
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
312
313
\psline[linecolor=blue]
314
(0.01999999999999999999999999999,0.003101264740186206637312896550)
315
(0.03999999999999999999999999999,0.006317998635585359010520157298)
316
(0.05999999999999999999999999999,0.009648390613582035143876795382)
317
(0.07999999999999999999999999999,0.01309056743894639526758693902)
318
(0.09999999999999999999999999999,0.01664259775815171562737670673)
319
(0.1199999999999999999999999999,0.02030249606158196955588582895)
320
(0.1399999999999999999999999999,0.02406822656114986486678020058)
321
(0.1599999999999999999999999999,0.02793770698113390008488216054)
322
(0.1799999999999999999999999999,0.03190881226031851732791999424)
323
(0.1999999999999999999999999999,0.03597937816378450422190788768)
324
(0.2199999999999999999999999999,0.04014720480294764495547411710)
325
(0.2399999999999999999999999999,0.04441006006268247008174737777)
326
(0.2599999999999999999999999999,0.04876568293459504765722696008)
327
(0.2799999999999999999999999999,0.05321178675572434797148408770)
328
(0.2999999999999999999999999999,0.05774606235215606370565235326)
329
(0.3199999999999999999999999999,0.06236618108722614873373364912)
330
(0.3399999999999999999999999999,0.06706979781417403114546156048)
331
(0.3599999999999999999999999999,0.07185455373327774471151462459)
332
(0.3799999999999999999999999999,0.07671807915366539815613898529)
333
(0.3999999999999999999999999999,0.08165799616014975732787858523)
334
(0.4199999999999999999999999999,0.08667192118557554858877807831)
335
(0.4399999999999999999999999999,0.09175746748930270129993406429)
336
(0.4599999999999999999999999999,0.09691224754257343301497348922)
337
(0.4799999999999999999999999999,0.1021338753216271429622334963)
338
(0.5000000000000000000000000000,0.1074199685095348171181575401)
339
(0.5199999999999999999999999999,0.1127681506078243599280920394)
340
(0.5399999999999999999999999999,0.1181760529590602499177597053)
341
(0.5599999999999999999999999999,0.1236413166816254629952563355)
342
(0.5799999999999999999999999999,0.1291615945180310091027382309)
343
(0.5999999999999999999999999999,0.1347345525981489724799498131)
344
(0.6199999999999999999999999999,0.1403578721188289166527752088)
345
(0.6400000000000000000000000000,0.1460292509414151915258882451)
346
(0.6600000000000000000000000000,0.1517464051087343361083354480)
347
(0.6800000000000000000000000000,0.1575070702831676758842033156)
348
(0.6999999999999999999999999999,0.1633090031074646328027749059)
349
(0.7200000000000000000000000000,0.1691499824899874568943655286)
350
(0.7400000000000000000000000000,0.1750278108161083044371831758)
351
(0.7600000000000000000000000000,0.1809403150875050752647980969)
352
(0.7800000000000000000000000000,0.1868853479911234219520755665)
353
(0.8000000000000000000000000000,0.1928607888995890907382271723)
354
(0.9000000000000000000000000000,0.2231218718049072360431784469)
355
(0.9200000000000000000000000000,0.2292368315756434886674200518)
356
(0.9400000000000000000000000000,0.2353681726309863443366734317)
357
(0.9600000000000000000000000000,0.2415140162133881652873199609)
358
(0.9800000000000000000000000000,0.2476725164678227936375651700)
359
(1.000000000000000000000000000,0.2538418608559106843377589237)
360
(1.020000000000000000000000000,0.2600202705212244228916944039)
361
(1.040000000000000000000000000,0.2662060006075045454652215771)
362
(1.060000000000000000000000000,0.2723973405314954285264279786)
363
(1.080000000000000000000000000,0.2785926142120892150770945963)
364
(1.100000000000000000000000000,0.2847901802574424333852867180)
365
(1.120000000000000000000000000,0.2909884321117052745610784846)
366
(1.140000000000000000000000000,0.2971857981629775541935773501)
367
(1.160000000000000000000000000,0.3033807418140783117843375717)
368
(1.180000000000000000000000000,0.3095717615176879155258869272)
369
(1.200000000000000000000000000,0.3157573907773925493281230058)
370
(1.220000000000000000000000000,0.3219361981161311688530138610)
371
(1.240000000000000000000000000,0.3281067870135145234967718814)
372
(1.260000000000000000000000000,0.3342677958134547465668457047)
373
(1.280000000000000000000000000,0.3404178976035124062765901834)
374
(1.300000000000000000000000000,0.3465558000673358708305202879)
375
(1.320000000000000000000000000,0.3526802453115354524147138581)
376
(1.340000000000000000000000000,0.3587900096683021335081805438)
377
(1.360000000000000000000000000,0.3648839034750478164517818140)
378
(1.380000000000000000000000000,0.3709607708323110413441686084)
379
(1.400000000000000000000000000,0.3770194893411390517445867594)
380
(1.420000000000000000000000000,0.3830589698211240121276578523)
381
(1.440000000000000000000000000,0.3890781560102381515827456359)
382
(1.460000000000000000000000000,0.3950760242475796772942255687)
383
(1.480000000000000000000000000,0.4010515831401085178143834242)
384
(1.500000000000000000000000000,0.4070038732144183656379131112)
385
(1.520000000000000000000000000,0.4129319665545591334819198790)
386
(1.540000000000000000000000000,0.4188349664268918582584391695)
387
(1.560000000000000000000000000,0.4247120068929263173294255750)
388
(1.580000000000000000000000000,0.4305622524110601967539437572)
389
(1.600000000000000000000000000,0.4363848974281076016577836211)
390
(1.620000000000000000000000000,0.4421791659614740527666030402)
391
(1.640000000000000000000000000,0.4479443111728048962566232025)
392
(1.660000000000000000000000000,0.4536796149339042897388580598)
393
(1.680000000000000000000000000,0.4593843873856926364966986941)
394
(1.700000000000000000000000000,0.4650579664909415419887042984)
395
(1.720000000000000000000000000,0.4706997175814970780131096773)
396
(1.740000000000000000000000000,0.4763090329006743757721108757)
397
(1.760000000000000000000000000,0.4818853311414793424952567260)
398
(1.780000000000000000000000000,0.4874280569812866186593691612)
399
(1.800000000000000000000000000,0.4929366806135767739015004303)
400
(1.820000000000000000000000000,0.4984106972773101876224958236)
401
(1.840000000000000000000000000,0.5038496267844900817063886842)
402
(1.860000000000000000000000000,0.5092530130464427730272425984)
403
(1.880000000000000000000000000,0.5146204235993193964607717449)
404
(1.900000000000000000000000000,0.5199514491293001177102494855)
405
(1.920000000000000000000000000,0.5252457029979592109837511789)
406
(1.940000000000000000000000000,0.5305028207682273199257343410)
407
(1.960000000000000000000000000,0.5357224597313657506972181836)
408
(1.980000000000000000000000000,0.5409042984353467622530070890)
409
(2.000000000000000000000000000,0.5460480362150135183341266606)
410
(2.020000000000000000000000000,0.5511533927243736453081566071)
411
(2.040000000000000000000000000,0.5562201074713611958144053871)
412
(2.060000000000000000000000000,0.5612479393553832455620974051)
413
(2.080000000000000000000000000,0.5662366662079493442915335815)
414
(2.100000000000000000000000000,0.5711860843366645959422388496)
415
(2.120000000000000000000000000,0.5760960080728502510276125755)
416
(2.140000000000000000000000000,0.5809662693230393491363441623)
417
(2.160000000000000000000000000,0.5857967171245791439501569968)
418
(2.180000000000000000000000000,0.5905872172055567693508898726)
419
(2.200000000000000000000000000,0.5953376515492498548760647386)
420
(2.220000000000000000000000000,0.6000479179632895634210866123)
421
(2.240000000000000000000000000,0.6047179296537097948265514854)
422
(2.260000000000000000000000000,0.6093476148040430667123888580)
423
(2.280000000000000000000000000,0.6139369161596108392744017913)
424
(2.300000000000000000000000000,0.6184857906171437841881210602)
425
(2.320000000000000000000000000,0.6229942088198556995414380047)
426
(2.340000000000000000000000000,0.6274621547580834329674692040)
427
(2.360000000000000000000000000,0.6318896253755942838795815015)
428
(2.380000000000000000000000000,0.6362766301816519028340103818)
429
(2.400000000000000000000000000,0.6406231908689216814032945376)
430
(2.420000000000000000000000000,0.6449293409372870193275923421)
431
(2.440000000000000000000000000,0.6491951253236386568835529626)
432
(2.460000000000000000000000000,0.6534206000376904591245617403)
433
(2.480000000000000000000000000,0.6576058318038666246625702406)
434
(2.500000000000000000000000000,0.6617508977092972547656467778)
435
(2.520000000000000000000000000,0.6658558848579515485622246546)
436
(2.540000000000000000000000000,0.6699208900309305769526543186)
437
(2.560000000000000000000000000,0.6739460193529346213787868396)
438
(2.580000000000000000000000000,0.6779313879649134339208017454)
439
(2.600000000000000000000000000,0.6818771197029014723967530728)
440
(2.620000000000000000000000000,0.6857833467830341784557455792)
441
(2.640000000000000000000000000,0.6896502094927356884132393584)
442
(2.660000000000000000000000000,0.6934778558880629862300000093)
443
(2.680000000000000000000000000,0.6972664414971864161662130006)
444
(2.700000000000000000000000000,0.7010161290299816599662799211)
445
(2.720000000000000000000000000,0.7047270880937037408067754255)
446
(2.740000000000000000000000000,0.7083994949147093346766878860)
447
(2.760000000000000000000000000,0.7120335320661896405150362043)
448
(2.780000000000000000000000000,0.7156293882018722746233703375)
449
(2.800000000000000000000000000,0.7191872577956471040781049612)
450
(2.820000000000000000000000000,0.7227073408870676097335862062)
451
(2.840000000000000000000000000,0.7261898428326762637425613552)
452
(2.860000000000000000000000000,0.7296349740630995113078967550)
453
(2.880000000000000000000000000,0.7330429498458552537718236202)
454
(2.900000000000000000000000000,0.7364139900538132324743725395)
455
(2.920000000000000000000000000,0.7397483189392464025736734233)
456
(2.940000000000000000000000000,0.7430461649134092558959284015)
457
(2.960000000000000000000000000,0.7463077603315770947268239419)
458
(2.980000000000000000000000000,0.7495333412834784672999949319)
459
(3.000000000000000000000000000,0.7527231473890513437891022943)
460
461
\rput(3.77,.75){$[0,-1,1,-10,-20]$}
462
\pscircle[linecolor=red](1,0){0.1}
463
\end{pspicture}
464
\caption{A graph of the $L$-series for $[0,-1,1,-10,-20]$}\label{rank11}
465
\end{center}
466
\end{figure}
467
The circle on the graph is drawn around the point (1,0), which is the critical point with respect to the BSD conjecture. It is critical in the sense that at this point the $L$-series should have the same order of vanishing as the rank of the elliptic curve. On this graph note that the graph does not appear to pass through that point. This would indicate that the rank of the $L$-series at 1 is zero. However, we also know that the rank of this curve is zero. Hence this graph agrees with the BSD conjecture.
468
469
Using our data we can compare the $L$-series for various curves of rank 1. This is shown in Figure~\ref{curvesof1}. The curves show in this graph are $[0,0,1,-1,0]$ with conductor 37, $[0,1,1,0,0]$ with conductor 43, $[1,-1,1,0,0]$ with conductor 53, and $[0,-1,1,-2,2]$ with conductor 57.
470
\begin{figure}
471
\begin{center}
472
\psset{unit=.9in}
473
\begin{pspicture}(-0.5,-1.5)(3,1.5)
474
\psgrid[gridcolor=gray]
475
476
% axes
477
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
478
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
479
480
\psline[linecolor=blue]
481
(0.01999999999999999999999999999,-0.007040682645273880424414896490)
482
(0.03999999999999999999999999999,-0.01384879340673429562897025802)
483
(0.05999999999999999999999999999,-0.02041151906125685194786353569)
484
(0.07999999999999999999999999999,-0.02671717089639622967868548846)
485
(0.09999999999999999999999999999,-0.03275514877468835667228254248)
486
(0.1199999999999999999999999999,-0.03851590459515947560090356606)
487
(0.1399999999999999999999999999,-0.04399090531018095579842531960)
488
(0.1599999999999999999999999999,-0.04917259564426959043634517837)
489
(0.1799999999999999999999999999,-0.05405436065038169549825383245)
490
(0.1999999999999999999999999999,-0.05863048822868273853342317371)
491
(0.2199999999999999999999999999,-0.06289613172268792282287897432)
492
(0.2399999999999999999999999999,-0.06684727269805707901875306754)
493
(0.2599999999999999999999999999,-0.07048068400018198510490385939)
494
(0.2799999999999999999999999999,-0.07379389317801729705702480679)
495
(0.2999999999999999999999999999,-0.07678514635336806772315486764)
496
(0.3199999999999999999999999999,-0.07945337260704693367480853604)
497
(0.3399999999999999999999999999,-0.08179814894594129630869963056)
498
(0.3599999999999999999999999999,-0.08381966590807343598837635362)
499
(0.3799999999999999999999999999,-0.08551869385618219352561940146)
500
(0.3999999999999999999999999999,-0.08689655000419094575245607525)
501
(0.4199999999999999999999999999,-0.08795506621514009498707863136)
502
(0.4399999999999999999999999999,-0.08869655760373996378059430584)
503
(0.4599999999999999999999999999,-0.08912379197162846543221041580)
504
(0.4799999999999999999999999999,-0.08923996009868375692678919869)
505
(0.5000000000000000000000000000,-0.08904864690933180697948009138)
506
(0.5199999999999999999999999999,-0.08855380352868900012719369984)
507
(0.5399999999999999999999999999,-0.08775972023957721056568979875)
508
(0.5599999999999999999999999999,-0.08667100034793001370606707268)
509
(0.5799999999999999999999999999,-0.08529253496086083061378244649)
510
(0.5999999999999999999999999999,-0.08362947867867400124606939994)
511
(0.6199999999999999999999999999,-0.08168722619935547704606842491)
512
(0.6400000000000000000000000000,-0.07947138983156869830977112899)
513
(0.6600000000000000000000000000,-0.07698777790989125167415357936)
514
(0.6800000000000000000000000000,-0.07424237410394737051996840888)
515
(0.6999999999999999999999999999,-0.07124131761120885189838824192)
516
(0.7200000000000000000000000000,-0.06799088422154145995628282789)
517
(0.7400000000000000000000000000,-0.06449746824005465646627421466)
518
(0.7600000000000000000000000000,-0.06076756525345918698261559700)
519
(0.7800000000000000000000000000,-0.05680775572393965919941740744)
520
(0.8000000000000000000000000000,-0.05262468939349814449812226300)
521
(0.8200000000000000000000000000,-0.04822507048081074570172472194)
522
(0.8400000000000000000000000000,-0.04361564365185310007207290383)
523
(0.8600000000000000000000000000,-0.03880318074488438664428006043)
524
(0.8800000000000000000000000000,-0.03379446822982440208845599349)
525
(0.9000000000000000000000000000,-0.02859629538160683747275401029)
526
(0.9200000000000000000000000000,-0.02321544314673655974290269603)
527
(0.9400000000000000000000000000,-0.01765867368201235289112268235)
528
(0.9600000000000000000000000000,-0.01193272054419242079363246703)
529
(0.9800000000000000000000000000,-0.006044279509271545062101302395)
530
(1.000000000000000000000000000,0)
531
(1.020000000000000000000000000,0.006193522899700657948953702821)
532
(1.040000000000000000000000000,0.01252975586468649046170633195)
533
(1.060000000000000000000000000,0.01900223420402991510662878509)
534
(1.080000000000000000000000000,0.02560456821288883018732881229)
535
(1.100000000000000000000000000,0.03233044902151853285873105062)
536
(1.120000000000000000000000000,0.03917365396187151515130066257)
537
(1.140000000000000000000000000,0.04612805147192782966336624274)
538
(1.160000000000000000000000000,0.05318760555756510911394608809)
539
(1.180000000000000000000000000,0.06034637983141611780977478710)
540
(1.200000000000000000000000000,0.06759854114777629370830779932)
541
(1.220000000000000000000000000,0.07493836285221727134548049211)
542
(1.240000000000000000000000000,0.08236022766413781101477395164)
543
(1.260000000000000000000000000,0.08985863021004364857227589662)
544
(1.280000000000000000000000000,0.09742817922489508043562107831)
545
(1.300000000000000000000000000,0.1050635994383979831195623428)
546
(1.320000000000000000000000000,0.1127597331626426342750000402)
547
(1.340000000000000000000000000,0.1205115415970171844667067337)
548
(1.360000000000000000000000000,0.1283141058658407995725197568)
549
(1.380000000000000000000000000,0.1361626278036770765653872408)
550
(1.400000000000000000000000000,0.1440524305028029124532985236)
551
(1.420000000000000000000000000,0.1519789586368230249427400150)
552
(1.440000000000000000000000000,0.1599377785739371047713213740)
553
(1.460000000000000000000000000,0.1679245782928863247835066101)
554
(1.480000000000000000000000000,0.1759351671141297281229851813)
555
(1.500000000000000000000000000,0.1839654752583298497321186624)
556
(1.520000000000000000000000000,0.1920115532437616743525970716)
557
(1.540000000000000000000000000,0.2000695711338004895487135877)
558
(1.560000000000000000000000000,0.2081358176451930554189518053)
559
(1.580000000000000000000000000,0.2162066991273734031110841806)
560
(1.600000000000000000000000000,0.2242787384226500348843475940)
561
(1.620000000000000000000000000,0.2323485736166658006761623977)
562
(1.640000000000000000000000000,0.2404129566881156747794000563)
563
(1.660000000000000000000000000,0.2484687520663013942874635967)
564
(1.680000000000000000000000000,0.2565129351047057339548403668)
565
(1.700000000000000000000000000,0.2645425904783833123832212255)
566
(1.720000000000000000000000000,0.2725549105125894351203245871)
567
(1.740000000000000000000000000,0.2805471934497037191070854782)
568
(1.760000000000000000000000000,0.2885168416611512059026883501)
569
(1.780000000000000000000000000,0.2964613598106804158541689015)
570
(1.800000000000000000000000000,0.3043783529750253442992429567)
571
(1.820000000000000000000000000,0.3122655247276567443190829145)
572
(1.840000000000000000000000000,0.3201206751910171395556735942)
573
(1.860000000000000000000000000,0.3279416990623337996760249098)
574
(1.880000000000000000000000000,0.3357265836178143006713637220)
575
(1.900000000000000000000000000,0.3434734066997501711381013130)
576
(1.920000000000000000000000000,0.3511803346907853633889956286)
577
(1.940000000000000000000000000,0.3588456204793477367406288223)
578
(1.960000000000000000000000000,0.3664676014199932362853021491)
579
(1.980000000000000000000000000,0.3740446972921738169877932115)
580
(2.000000000000000000000000000,0.3815754082607112112937104095)
581
(2.020000000000000000000000000,0.3890583128410391695598633238)
582
(2.040000000000000000000000000,0.3964920658720666086723007354)
583
(2.060000000000000000000000000,0.4038753964993129699142493503)
584
(2.080000000000000000000000000,0.4112071061707747910139988429)
585
(2.100000000000000000000000000,0.4184860666477988128610003466)
586
(2.120000000000000000000000000,0.4257112180330616382145949304)
587
(2.140000000000000000000000000,0.4328815668175888044935202098)
588
(2.160000000000000000000000000,0.4399961839485868900507079621)
589
(2.180000000000000000000000000,0.4470542029197107066446098109)
590
(2.200000000000000000000000000,0.4540548178852435031314417493)
591
(2.220000000000000000000000000,0.4609972817995311800149907190)
592
(2.240000000000000000000000000,0.4678809045828815556061141898)
593
(2.260000000000000000000000000,0.4747050513150164978537264130)
594
(2.280000000000000000000000000,0.4814691404570480091101373545)
595
(2.300000000000000000000000000,0.4881726421028388943567929956)
596
(2.320000000000000000000000000,0.4948150762605042298041400070)
597
(2.340000000000000000000000000,0.5013960111647112546180296503)
598
(2.360000000000000000000000000,0.5079150616203423137352231886)
599
(2.380000000000000000000000000,0.5143718873779978681164907961)
600
(2.400000000000000000000000000,0.5207661915417341482886693073)
601
(2.420000000000000000000000000,0.5270977190093525499433942425)
602
(2.440000000000000000000000000,0.5333662549454851535285664326)
603
(2.460000000000000000000000000,0.5395716232876525947265009191)
604
(2.480000000000000000000000000,0.5457136852854067258264170737)
605
(2.500000000000000000000000000,0.5517923380726109005679500842)
606
(2.520000000000000000000000000,0.5578075132728551033706124653)
607
(2.540000000000000000000000000,0.5637591756379513493598187888)
608
(2.560000000000000000000000000,0.5696473217194066307559695957)
609
(2.580000000000000000000000000,0.5754719785727260096488165877)
610
(2.600000000000000000000000000,0.5812332024943570937173917601)
611
(2.620000000000000000000000000,0.5869310777910489219925825373)
612
(2.640000000000000000000000000,0.5925657155813630793228431299)
613
(2.660000000000000000000000000,0.5981372526290425028982153542)
614
(2.680000000000000000000000000,0.6036458502079137991495723328)
615
(2.700000000000000000000000000,0.6090916929979718166801361655)
616
(2.720000000000000000000000000,0.6144749880122705876273751187)
617
(2.740000000000000000000000000,0.6197959635542224278568417655)
618
(2.760000000000000000000000000,0.6250548682048868522766910655)
619
(2.780000000000000000000000000,0.6302519698398128966284810703)
620
(2.800000000000000000000000000,0.6353875546749823272365581299)
621
(2.820000000000000000000000000,0.6404619263413869557542058624)
622
(2.840000000000000000000000000,0.6454754049877607516894869008)
623
(2.860000000000000000000000000,0.6504283264109765604753116705)
624
(2.880000000000000000000000000,0.6553210412136078922982997797)
625
(2.900000000000000000000000000,0.6601539139881483541278100795)
626
(2.920000000000000000000000000,0.6649273225273747656668244570)
627
(2.940000000000000000000000000,0.6696416570603347444156881135)
628
(2.960000000000000000000000000,0.6742973195134354845820148258)
629
(2.980000000000000000000000000,0.6788947227961075117071180513)
630
(3.000000000000000000000000000,0.6834342901105152956599508217)
631
632
\psline[linecolor=green]
633
(0.01999999999999999999999999999,-0.008639208544790906071799184600)
634
(0.03999999999999999999999999999,-0.01695730708895535366918586418)
635
(0.05999999999999999999999999999,-0.02494082955571175614366979694)
636
(0.07999999999999999999999999999,-0.03257778442516353460060332836)
637
(0.09999999999999999999999999999,-0.03985759436432216024632235538)
638
(0.1199999999999999999999999999,-0.04677103599281271869156288637)
639
(0.1399999999999999999999999999,-0.05331017997207616552373419999)
640
(0.1599999999999999999999999999,-0.05946833158843696287688175590)
641
(0.1799999999999999999999999999,-0.06523997198391352447125254229)
642
(0.1999999999999999999999999999,-0.07062070017308674859534011580)
643
(0.2199999999999999999999999999,-0.07560717596968229927940639786)
644
(0.2399999999999999999999999999,-0.08019706393273741532477075180)
645
(0.2599999999999999999999999999,-0.08438897842928414333959093130)
646
(0.2799999999999999999999999999,-0.08818242989835854166578812809)
647
(0.2999999999999999999999999999,-0.09157777238980961155723021506)
648
(0.3199999999999999999999999999,-0.09457615244080218035535756517)
649
(0.3399999999999999999999999999,-0.09717945934305423402232858927)
650
(0.3599999999999999999999999999,-0.09939027684469081556336661711)
651
(0.3799999999999999999999999999,-0.1012118363221032464172534268)
652
(0.3999999999999999999999999999,-0.1026479714493440173723752579)
653
(0.4199999999999999999999999999,-0.1037030743853345203998051017)
654
(0.4399999999999999999999999999,-0.1043820534924855913435603823)
655
(0.4599999999999999999999999999,-0.1046902925942008769634240957)
656
(0.4799999999999999999999999999,-0.1046336117731222024894886764)
657
(0.5000000000000000000000000000,-0.1042182297068569341806201178)
658
(0.5199999999999999999999999999,-0.1034507275332730546651753436)
659
(0.5399999999999999999999999999,-0.1023380142332323007625026531)
660
(0.5599999999999999999999999999,-0.1008872935148300459470240305)
661
(0.5799999999999999999999999999,-0.09910603217979824944926878350)
662
(0.5999999999999999999999999999,-0.09700192994968118313837821093)
663
(0.6199999999999999999999999999,-0.09458289072669007720426825721)
664
(0.6400000000000000000000000000,-0.09185699526176044615940873735)
665
(0.6600000000000000000000000000,-0.08883247520025368049548403663)
666
(0.6800000000000000000000000000,-0.08551768847394239194751208751)
667
(0.6999999999999999999999999999,-0.08192109600637771625455013134)
668
(0.7200000000000000000000000000,-0.07805123969743789296711172043)
669
(0.7400000000000000000000000000,-0.07391672165178338542081691590)
670
(0.7600000000000000000000000000,-0.06952618461507783272655026135)
671
(0.7800000000000000000000000000,-0.06488829358116031087046388401)
672
(0.8000000000000000000000000000,-0.06001171853285759030155842735)
673
(0.8200000000000000000000000000,-0.05490511827879095983535031075)
674
(0.8400000000000000000000000000,-0.04957712534834714705906533986)
675
(0.8600000000000000000000000000,-0.04403633190693404690114308651)
676
(0.8800000000000000000000000000,-0.03829127665371723123428925841)
677
(0.9000000000000000000000000000,-0.03235043266422110449948981417)
678
(0.9200000000000000000000000000,-0.02622219614046831374461425287)
679
(0.9400000000000000000000000000,-0.01991487603171248195901849325)
680
(0.9600000000000000000000000000,-0.01343668448928299834021055606)
681
(0.9800000000000000000000000000,-0.006795728119597552585055158310)
682
(1.000000000000000000000000000,0)
683
(1.020000000000000000000000000,0.006942627577259803208644215643)
684
(1.040000000000000000000000000,0.01402440966687861781192303062)
685
(1.060000000000000000000000000,0.02123773457140726038402721734)
686
(1.080000000000000000000000000,0.02857512911364634497343689641)
687
(1.100000000000000000000000000,0.03602926331214392555958645615)
688
(1.120000000000000000000000000,0.04359295449056315405028036313)
689
(1.140000000000000000000000000,0.05125917085082197339855406509)
690
(1.160000000000000000000000000,0.05902103453902475511836846861)
691
(1.180000000000000000000000000,0.06687182423231189221021759028)
692
(1.200000000000000000000000000,0.07480497727385154974953788711)
693
(1.220000000000000000000000000,0.08281409138229160335492603438)
694
(1.240000000000000000000000000,0.09089292596108249614456823405)
695
(1.260000000000000000000000000,0.09903540303217625990659917374)
696
(1.280000000000000000000000000,0.1072356078177059423799933465)
697
(1.300000000000000000000000000,0.1154877889923555668903819291)
698
(1.320000000000000000000000000,0.1237863586282456871270627039)
699
(1.340000000000000000000000000,0.1321258918532855249271197576)
700
(1.360000000000000000000000000,0.1405011262430813159929779263)
701
(1.380000000000000000000000000,0.1489069609656433621738921841)
702
(1.400000000000000000000000000,0.1573384556973027386513402085)
703
(1.420000000000000000000000000,0.1657908293274337969908656261)
704
(1.440000000000000000000000000,0.1742594584687815472427639890)
705
(1.460000000000000000000000000,0.1827398757894145521781497115)
706
(1.480000000000000000000000000,0.1912277681815648448636444773)
707
(1.500000000000000000000000000,0.1997189747818771805429345802)
708
(1.520000000000000000000000000,0.2082094848568711314610666491)
709
(1.540000000000000000000000000,0.2166954355667214972837208656)
710
(1.560000000000000000000000000,0.2251731096197855036200398134)
711
(1.580000000000000000000000000,0.2336389328296494757415873231)
712
(1.600000000000000000000000000,0.2420894715858332000535891049)
713
(1.620000000000000000000000000,0.2505214302486770430364515045)
714
(1.640000000000000000000000000,0.2589316484783450386767504132)
715
(1.660000000000000000000000000,0.2673170985073064714379246564)
716
(1.680000000000000000000000000,0.2756748823651088074382057858)
717
(1.700000000000000000000000000,0.2840022290637259465530739246)
718
(1.720000000000000000000000000,0.2922964917512574228266345327)
719
(1.740000000000000000000000000,0.3005551448412660703755286146)
720
(1.760000000000000000000000000,0.3087757811245734623954130540)
721
(1.780000000000000000000000000,0.3169561088698837567353681184)
722
(1.800000000000000000000000000,0.3250939489191770509163106755)
723
(1.820000000000000000000000000,0.3331872317834025476128051860)
724
(1.840000000000000000000000000,0.3412339947436093241702040174)
725
(1.860000000000000000000000000,0.3492323789622778360532961278)
726
(1.880000000000000000000000000,0.3571806266092580002052557435)
727
(1.900000000000000000000000000,0.3650770780063793254515214242)
728
(1.920000000000000000000000000,0.3729201687944746004383791723)
729
(1.940000000000000000000000000,0.3807084271262506263625438801)
730
(1.960000000000000000000000000,0.3884404708881468993028218036)
731
(1.980000000000000000000000000,0.3961150049540455107443515831)
732
(2.000000000000000000000000000,0.4037308184734323501075578474)
733
(2.020000000000000000000000000,0.4112867821963604663163287638)
734
(2.040000000000000000000000000,0.4187818458373306859704221016)
735
(2.060000000000000000000000000,0.4262150354799818068479615650)
736
(2.080000000000000000000000000,0.4335854510242724057257426789)
737
(2.100000000000000000000000000,0.4408922636776380434968657688)
738
(2.120000000000000000000000000,0.4481347134914209499690567205)
739
(2.140000000000000000000000000,0.4553121069436936650660289074)
740
(2.160000000000000000000000000,0.4624238145694331504809859457)
741
(2.180000000000000000000000000,0.4694692686388471233243711628)
742
(2.200000000000000000000000000,0.4764479608845093677828245209)
743
(2.220000000000000000000000000,0.4833594402778251291481300718)
744
(2.240000000000000000000000000,0.4902033108552209740982226119)
745
(2.260000000000000000000000000,0.4969792295943353098539720162)
746
(2.280000000000000000000000000,0.5036869043403757017864417879)
747
(2.300000000000000000000000000,0.5103260917827068343300664352)
748
(2.320000000000000000000000000,0.5168965954816380550416206698)
749
(2.340000000000000000000000000,0.5233982639452915690374328534)
750
(2.360000000000000000000000000,0.5298309887563511649114931408)
751
(2.380000000000000000000000000,0.5361947027484165190151433141)
752
(2.400000000000000000000000000,0.5424893782316193194166756350)
753
(2.420000000000000000000000000,0.5487150252670943619590274819)
754
(2.440000000000000000000000000,0.5548716899898410977503608247)
755
(2.460000000000000000000000000,0.5609594529794585643379261810)
756
(2.480000000000000000000000000,0.5669784276781889327936568856)
757
(2.500000000000000000000000000,0.5729287588556617817586303980)
758
(2.520000000000000000000000000,0.5788106211196924094621048830)
759
(2.540000000000000000000000000,0.5846242174724527684906711546)
760
(2.560000000000000000000000000,0.5903697779113027183957436417)
761
(2.580000000000000000000000000,0.5960475580735420107518910716)
762
(2.600000000000000000000000000,0.6016578379243195323378500536)
763
(2.620000000000000000000000000,0.6072009204869156264545250877)
764
(2.640000000000000000000000000,0.6126771306145955909461063054)
765
(2.660000000000000000000000000,0.6180868138032175241048335315)
766
(2.680000000000000000000000000,0.6234303350437653748422042464)
767
(2.700000000000000000000000000,0.6287080777139681782396512775)
768
(2.720000000000000000000000000,0.6339204425081588569460400727)
769
(2.740000000000000000000000000,0.6390678464045204858607283658)
770
(2.760000000000000000000000000,0.6441507216688644027603692392)
771
(2.780000000000000000000000000,0.6491695148940828590064178928)
772
(2.800000000000000000000000000,0.6541246860744189073441972571)
773
(2.820000000000000000000000000,0.6590167077136977900802617444)
774
(2.840000000000000000000000000,0.6638460639666670992302797991)
775
(2.860000000000000000000000000,0.6686132498125973155662130733)
776
(2.880000000000000000000000000,0.6733187702602998869916214168)
777
(2.900000000000000000000000000,0.6779631395837266753622035110)
778
(2.920000000000000000000000000,0.6825468805873222874282555780)
779
(2.940000000000000000000000000,0.6870705239003094181189093130)
780
(2.960000000000000000000000000,0.6915346072990967862324347602)
781
(2.980000000000000000000000000,0.6959396750570094520469511368)
782
(3.000000000000000000000000000,0.7002862773205521964996432703)
783
784
\psline[linecolor=cyan]
785
(0.01999999999999999999999999999,-0.01232359015235256344883482079)
786
(0.03999999999999999999999999999,-0.02412446588889267226889696308)
787
(0.05999999999999999999999999999,-0.03538785546259972017085210186)
788
(0.07999999999999999999999999999,-0.04610122171142140960027556198)
789
(0.09999999999999999999999999999,-0.05625414612889305476593374889)
790
(0.1199999999999999999999999999,-0.06583821507745232526592669292)
791
(0.1399999999999999999999999999,-0.07484690836601214282431585346)
792
(0.1599999999999999999999999999,-0.08327549038389164433002770289)
793
(0.1799999999999999999999999999,-0.09112090395581891063990775832)
794
(0.1999999999999999999999999999,-0.09838166705731753231806887441)
795
(0.2199999999999999999999999999,-0.1050577725062806215766963638)
796
(0.2399999999999999999999999999,-0.1111505907248301297181797450)
797
(0.2599999999999999999999999999,-0.1166627756455671362226616035)
798
(0.2799999999999999999999999999,-0.1215981738179525732772071301)
799
(0.2999999999999999999999999999,-0.1259617367537319098133044645)
800
(0.3199999999999999999999999999,-0.1297594365349479626809820289)
801
(0.3399999999999999999999999999,-0.1329981846940917756875675838)
802
(0.3599999999999999999999999999,-0.1356857543632433271494220127)
803
(0.3799999999999999999999999999,-0.1378307056775750987602650611)
804
(0.3999999999999999999999999999,-0.1394423144082582479447360931)
805
(0.4199999999999999999999999999,-0.1405305037905519052486047536)
806
(0.4399999999999999999999999999,-0.1411057795046022958142483213)
807
(0.4599999999999999999999999999,-0.1411791677591640117964890352)
808
(0.4799999999999999999999999999,-0.1407621564220176291592503077)
809
(0.5000000000000000000000000000,-0.1398666391352354898086445991)
810
(0.5199999999999999999999999999,-0.1385048623485830986520529738)
811
(0.5399999999999999999999999999,-0.1366893752001821460455378890)
812
(0.5599999999999999999999999999,-0.1344329821700502441424408516)
813
(0.5799999999999999999999999999,-0.1317486984292222538638358386)
814
(0.5999999999999999999999999999,-0.1286497078048013253793649442)
815
(0.6199999999999999999999999999,-0.1251493232794397222048280840)
816
(0.6400000000000000000000000000,-0.1212609499423678208356276114)
817
(0.6600000000000000000000000000,-0.1169980503081344090433836259)
818
(0.6800000000000000000000000000,-0.1123741119186548698998107512)
819
(0.6999999999999999999999999999,-0.1074026171439505716379021269)
820
(0.7200000000000000000000000000,-0.1020970150970694574844209198)
821
(0.7400000000000000000000000000,-0.09647069557907317294044895336)
822
(0.7600000000000000000000000000,-0.09053696497063078554082032181)
823
(0.7800000000000000000000000000,-0.08430902398764584500168189416)
824
(0.8000000000000000000000000000,-0.07779994721943661689699157130)
825
(0.8200000000000000000000000000,-0.07102266436926495364264487036)
826
(0.8400000000000000000000000000,-0.06398994311844525254987818457)
827
(0.8600000000000000000000000000,-0.05671437353684068077240087240)
828
(0.8800000000000000000000000000,-0.04920835396425021186137368979)
829
(0.9000000000000000000000000000,-0.04148407828898933586073228211)
830
(0.9200000000000000000000000000,-0.03355352455185324513545942580)
831
(0.9400000000000000000000000000,-0.02542844480560881367866008585)
832
(0.9600000000000000000000000000,-0.01712035616217694260754587130)
833
(0.9800000000000000000000000000,-0.008640532961727147351744217640)
834
(1.000000000000000000000000000,0)
835
(1.020000000000000000000000000,0.008790473247710383461527368054)
836
(1.040000000000000000000000000,0.01772037746535184861721081667)
837
(1.060000000000000000000000000,0.02677946745499151477369554376)
838
(1.080000000000000000000000000,0.03595776527094793837367008060)
839
(1.100000000000000000000000000,0.04524556258223097580311741242)
840
(1.120000000000000000000000000,0.05463342231423895403161554190)
841
(1.140000000000000000000000000,0.06411217961858217266811117088)
842
(1.160000000000000000000000000,0.07367294221784889028047124539)
843
(1.180000000000000000000000000,0.08330709017010925640526541660)
844
(1.200000000000000000000000000,0.09300627509596805215223969798)
845
(1.220000000000000000000000000,0.1027624189090319990230633688)
846
(1.240000000000000000000000000,0.1125677120887547168619144152)
847
(1.260000000000000000000000000,0.1224146115327646503143203931)
848
(1.280000000000000000000000000,0.1322958380239705288498892231)
849
(1.300000000000000000000000000,0.1422043733459768981827900296)
850
(1.320000000000000000000000000,0.1521334570786303409897488909)
851
(1.340000000000000000000000000,0.1620765831038562613878535040)
852
(1.360000000000000000000000000,0.1720274958503373259194113994)
853
(1.380000000000000000000000000,0.1819801863040283603914350844)
854
(1.400000000000000000000000000,0.1919288878099989885646329751)
855
(1.420000000000000000000000000,0.2018680716896446445111287890)
856
(1.440000000000000000000000000,0.2117924426959086828047897812)
857
(1.460000000000000000000000000,0.2216969343278128645649526068)
858
(1.480000000000000000000000000,0.2315767040243000735049581438)
859
(1.500000000000000000000000000,0.2414271282561511379420059340)
860
(1.520000000000000000000000000,0.2512437975335464039252520398)
861
(1.540000000000000000000000000,0.2610225113457014158280585906)
862
(1.560000000000000000000000000,0.2707592730479138148763828420)
863
(1.580000000000000000000000000,0.2804502847103143829228033395)
864
(1.600000000000000000000000000,0.2900919419416179884384106488)
865
(1.620000000000000000000000000,0.2996808287002189252462709042)
866
(1.640000000000000000000000000,0.3092137121040686137322529661)
867
(1.660000000000000000000000000,0.3186875372499106605784695479)
868
(1.680000000000000000000000000,0.3280994220516276157755483479)
869
(1.700000000000000000000000000,0.3374466521066741694480249982)
870
(1.720000000000000000000000000,0.3467266755988317227291668406)
871
(1.740000000000000000000000000,0.3559370982448179618029671132)
872
(1.760000000000000000000000000,0.3650756782916209715712216335)
873
(1.780000000000000000000000000,0.3741403215707992535870021122)
874
(1.800000000000000000000000000,0.3831290766153954739859680027)
875
(1.820000000000000000000000000,0.3920401298445515809873908567)
876
(1.840000000000000000000000000,0.4008718008203848294123967855)
877
(1.860000000000000000000000000,0.4096225375811869775566140150)
878
(1.880000000000000000000000000,0.4182909120545412432045890793)
879
(1.900000000000000000000000000,0.4268756155535123042206950524)
880
(1.920000000000000000000000000,0.4353754543586525109208487449)
881
(1.940000000000000000000000000,0.4437893453881813724331634040)
882
(1.960000000000000000000000000,0.4521163119583341434151694010)
883
(1.980000000000000000000000000,0.4603554796355378539101485825)
884
(2.000000000000000000000000000,0.4685060721817583052091355785)
885
(2.020000000000000000000000000,0.4765674075940683389882470898)
886
(2.040000000000000000000000000,0.4845388942392150463268382428)
887
(2.060000000000000000000000000,0.4924200270837105186072879247)
888
(2.080000000000000000000000000,0.5002103840197362857795868456)
889
(2.100000000000000000000000000,0.5079096222869348022129058038)
890
(2.120000000000000000000000000,0.5155174749899613207719614573)
891
(2.140000000000000000000000000,0.5230337477114853675076708079)
892
(2.160000000000000000000000000,0.5304583152201619492197161922)
893
(2.180000000000000000000000000,0.5377911182729377818203129670)
894
(2.200000000000000000000000000,0.5450321605109164371975939494)
895
(2.220000000000000000000000000,0.5521815054478776186687933016)
896
(2.240000000000000000000000000,0.5592392735504290684420794161)
897
(2.260000000000000000000000000,0.5662056394086641923741956030)
898
(2.280000000000000000000000000,0.5730808289961036940590570127)
899
(2.300000000000000000000000000,0.5798651170176147063927142158)
900
(2.320000000000000000000000000,0.5865588243439254862260160353)
901
(2.340000000000000000000000000,0.5931623155312871153792807958)
902
(2.360000000000000000000000000,0.5996759964247752741543796257)
903
(2.380000000000000000000000000,0.6061003118436744919907083444)
904
(2.400000000000000000000000000,0.6124357433473438292503969723)
905
(2.420000000000000000000000000,0.6186828070799262234519183900)
906
(2.440000000000000000000000000,0.6248420516922332850117966145)
907
(2.460000000000000000000000000,0.6309140563391127166085232349)
908
(2.480000000000000000000000000,0.6368994287505863433027398246)
909
(2.500000000000000000000000000,0.6427988033750325851798414952)
910
(2.520000000000000000000000000,0.6486128395926777084210612381)
911
(2.540000000000000000000000000,0.6543422199976550017615587321)
912
(2.560000000000000000000000000,0.6599876487468898094408347628)
913
(2.580000000000000000000000000,0.6655498499740707932282672384)
914
(2.600000000000000000000000000,0.6710295662669735964943208143)
915
(2.620000000000000000000000000,0.6764275572064119608200396978)
916
(2.640000000000000000000000000,0.6817445979651030344769970157)
917
(2.660000000000000000000000000,0.6869814779647478617386918134)
918
(2.680000000000000000000000000,0.6921389995896446165075226276)
919
(2.700000000000000000000000000,0.6972179769551708212581904844)
920
(2.720000000000000000000000000,0.7022192347294913642797055223)
921
(2.740000000000000000000000000,0.7071436070068713988815025067)
922
(2.760000000000000000000000000,0.7119919362309969940304274431)
923
(2.780000000000000000000000000,0.7167650721667315348287534282)
924
(2.800000000000000000000000000,0.7214638709187621824090677607)
925
(2.820000000000000000000000000,0.7260891939956180458114906948)
926
(2.840000000000000000000000000,0.7306419074175699528288746094)
927
(2.860000000000000000000000000,0.7351228808669507017693798919)
928
(2.880000000000000000000000000,0.7395329868794643097315978208)
929
(2.900000000000000000000000000,0.7438731000750829320158584216)
930
(2.920000000000000000000000000,0.7481440964271607065235009068)
931
(2.940000000000000000000000000,0.7523468525684246789282390927)
932
(2.960000000000000000000000000,0.7564822451325340988205483643)
933
(2.980000000000000000000000000,0.7605511501299306605878075618)
934
(3.000000000000000000000000000,0.7645544423567336186615549801)
935
936
\psline[linecolor=magenta]
937
(0.01999999999999999999999999999,-0.01244581056794672230484472646)
938
(0.03999999999999999999999999999,-0.02432984148580824889923158870)
939
(0.05999999999999999999999999999,-0.03563988114461296550223351770)
940
(0.07999999999999999999999999999,-0.04636601040627589032487973371)
941
(0.09999999999999999999999999999,-0.05650046832950686233575658099)
942
(0.1199999999999999999999999999,-0.06603752141479784844225823619)
943
(0.1399999999999999999999999999,-0.07497333654666454348009900467)
944
(0.1599999999999999999999999999,-0.08330585777944427693621113576)
945
(0.1799999999999999999999999999,-0.09103468708363809338270413785)
946
(0.1999999999999999999999999999,-0.09816096914289608802000686039)
947
(0.2199999999999999999999999999,-0.1046872802671508306939998951)
948
(0.2399999999999999999999999999,-0.1106175214649763511272253491)
949
(0.2599999999999999999999999999,-0.1159568156978654971205726674)
950
(0.2799999999999999999999999999,-0.1207114093206560275132035368)
951
(0.2999999999999999999999999999,-0.1248885776956789630128457810)
952
(0.3199999999999999999999999999,-0.1284965349532389271336012293)
953
(0.3399999999999999999999999999,-0.1315443478576570482850008148)
954
(0.3599999999999999999999999999,-0.1340418537262082647108018484)
955
(0.3799999999999999999999999999,-0.1359995823377666432747610608)
956
(0.3999999999999999999999999999,-0.1374286817587389396212432033)
957
(0.4199999999999999999999999999,-0.1383408480058267144918997640)
958
(0.4399999999999999999999999999,-0.1387482584582237492913855108)
959
(0.4599999999999999999999999999,-0.1386635089259453439695622057)
960
(0.4799999999999999999999999999,-0.1380995542760205401372918408)
961
(0.5000000000000000000000000000,-0.1370696525141826613604201331)
962
(0.5199999999999999999999999999,-0.1355873122163970428351237066)
963
(0.5399999999999999999999999999,-0.1336662432020005506342751506)
964
(0.5599999999999999999999999999,-0.1313203103383323507810046011)
965
(0.5799999999999999999999999999,-0.1285634903654499194183465908)
966
(0.5999999999999999999999999999,-0.1254098316287925619701699748)
967
(0.6199999999999999999999999999,-0.1218734166074242190579984800)
968
(0.6400000000000000000000000000,-0.1179683271257088547949838673)
969
(0.6600000000000000000000000000,-0.1137086121368991829259211395)
970
(0.6800000000000000000000000000,-0.1091082579681098524133485579)
971
(0.6999999999999999999999999999,-0.1041811609174593517708626584)
972
(0.7200000000000000000000000000,-0.09894110209576343807582234429)
973
(0.7400000000000000000000000000,-0.09340172440701213365653693937)
974
(0.7600000000000000000000000000,-0.08757651156393006018009475600)
975
(0.7800000000000000000000000000,-0.08147876903717628881743822321)
976
(0.8000000000000000000000000000,-0.07512160683915744016577357692)
977
(0.8200000000000000000000000000,-0.06851792404598108447403724282)
978
(0.8400000000000000000000000000,-0.06168039496374222293284165330)
979
(0.8600000000000000000000000000,-0.05462145684809234847165497345)
980
(0.8800000000000000000000000000,-0.04735329908886867678122884023)
981
(0.9000000000000000000000000000,-0.03988785377444269926125070274)
982
(0.9200000000000000000000000000,-0.03223678755336593805175378012)
983
(0.9400000000000000000000000000,-0.02441149471383188412426769156)
984
(0.9600000000000000000000000000,-0.01642309140442318887045462929)
985
(0.9800000000000000000000000000,-0.008282410922560194573676627870)
986
(1.000000000000000000000000000,0)
987
(1.020000000000000000000000000,0.008413883982355215037163819992)
988
(1.040000000000000000000000000,0.01694927491520454590426346098)
989
(1.060000000000000000000000000,0.02559649907917580720582942490)
990
(1.080000000000000000000000000,0.03434617551996943130967777580)
991
(1.100000000000000000000000000,0.04318921573681547302452605812)
992
(1.120000000000000000000000000,0.05211682273805902486703732843)
993
(1.140000000000000000000000000,0.06112048951506968038052236242)
994
(1.160000000000000000000000000,0.07019199698311633700280432568)
995
(1.180000000000000000000000000,0.07932341143536262851426721198)
996
(1.200000000000000000000000000,0.08850708155372307780703073625)
997
(1.220000000000000000000000000,0.09773563501797760387084340940)
998
(1.240000000000000000000000000,0.1070019747522737708240279441)
999
(1.260000000000000000000000000,0.1162992748459531705907299374)
1000
(1.280000000000000000000000000,0.1256209761835212341967801550)
1001
(1.300000000000000000000000000,0.1349607818165388674648738923)
1002
(1.320000000000000000000000000,0.1443126521082495859734594983)
1003
(1.340000000000000000000000000,0.1536707996798669779021319253)
1004
(1.360000000000000000000000000,0.1630296841856337940995227762)
1005
(1.380000000000000000000000000,0.1723840069420249687235496702)
1006
(1.400000000000000000000000000,0.1817287054348014276884134410)
1007
(1.420000000000000000000000000,0.1910589477260284869380718678)
1008
(1.440000000000000000000000000,0.2003701267816506661582141520)
1009
(1.460000000000000000000000000,0.2096578547387624014146225954)
1010
(1.480000000000000000000000000,0.2189179571303298744813979060)
1011
(1.500000000000000000000000000,0.2281464670838013335899238187)
1012
(1.520000000000000000000000000,0.2373396195087901265497446165)
1013
(1.540000000000000000000000000,0.2464938452878244042317144225)
1014
(1.560000000000000000000000000,0.2556057654830282302941523525)
1015
(1.580000000000000000000000000,0.2646721855705287625391926114)
1016
(1.600000000000000000000000000,0.2736900897133713350536233945)
1017
(1.620000000000000000000000000,0.2826566350827667329410320929)
1018
(1.640000000000000000000000000,0.2915691462365907687182194727)
1019
(1.660000000000000000000000000,0.3004251095632034963722200980)
1020
(1.680000000000000000000000000,0.3092221677978520973774051111)
1021
(1.700000000000000000000000000,0.3179581146181657176312524907)
1022
(1.720000000000000000000000000,0.3266308893245404193549014869)
1023
(1.740000000000000000000000000,0.3352385716105460558532938548)
1024
(1.760000000000000000000000000,0.3437793764278624267687783156)
1025
(1.780000000000000000000000000,0.3522516489496677069735920841)
1026
(1.800000000000000000000000000,0.3606538596358560795872304639)
1027
(1.820000000000000000000000000,0.3689845994029519979351126863)
1028
(1.840000000000000000000000000,0.3772425749011138493276960086)
1029
(1.860000000000000000000000000,0.3854266039001783357220130207)
1030
(1.880000000000000000000000000,0.3935356107862870083478858372)
1031
(1.900000000000000000000000000,0.4015686221702565276245083537)
1032
(1.920000000000000000000000000,0.4095247626085028462376449269)
1033
(1.940000000000000000000000000,0.4174032504370051606017727238)
1034
(1.960000000000000000000000000,0.4252033937184967215134499067)
1035
(1.980000000000000000000000000,0.4329245863027950651864099857)
1036
(2.000000000000000000000000000,0.4405663039999325968119333370)
1037
(2.020000000000000000000000000,0.4481281008655184551167455754)
1038
(2.040000000000000000000000000,0.4556096055975529816223615136)
1039
(2.060000000000000000000000000,0.4630105180437257342235085905)
1040
(2.080000000000000000000000000,0.4703306058180556907294996197)
1041
(2.100000000000000000000000000,0.4775697010255770005171462302)
1042
(2.120000000000000000000000000,0.4847276970936343239118619628)
1043
(2.140000000000000000000000000,0.4918045457082274570450581147)
1044
(2.160000000000000000000000000,0.4988002538537346266678789506)
1045
(2.180000000000000000000000000,0.5057148809542466498667039893)
1046
(2.200000000000000000000000000,0.5125485361146592250598353201)
1047
(2.220000000000000000000000000,0.5193013754595971312583241549)
1048
(2.240000000000000000000000000,0.5259735995681812803431685784)
1049
(2.260000000000000000000000000,0.5325654510025966486410137371)
1050
(2.280000000000000000000000000,0.5390772119283754033491673992)
1051
(2.300000000000000000000000000,0.5455092018242743665019615326)
1052
(2.320000000000000000000000000,0.5518617752795986892395184042)
1053
(2.340000000000000000000000000,0.5581353198768036408784568629)
1054
(2.360000000000000000000000000,0.5643302541571931818885110203)
1055
(2.380000000000000000000000000,0.5704470256675269497772066017)
1056
(2.400000000000000000000000000,0.5764861090853459345237313988)
1057
(2.420000000000000000000000000,0.5824480044208309768523209996)
1058
(2.440000000000000000000000000,0.5883332352930168372073727391)
1059
(2.460000000000000000000000000,0.5941423472781975311839781333)
1060
(2.480000000000000000000000000,0.5998759063283755091236636801)
1061
(2.500000000000000000000000000,0.6055344972576276985872725962)
1062
(2.520000000000000000000000000,0.6111187222942850765971454832)
1063
(2.540000000000000000000000000,0.6166291996968489641232796769)
1064
(2.560000000000000000000000000,0.6220665624315963295592970760)
1065
(2.580000000000000000000000000,0.6274314569098577622414611648)
1066
(2.600000000000000000000000000,0.6327245417829851618456578772)
1067
(2.620000000000000000000000000,0.6379464867930613333409862000)
1068
(2.640000000000000000000000000,0.6430979716774403459120593237)
1069
(2.660000000000000000000000000,0.6481796851252454900737272486)
1070
(2.680000000000000000000000000,0.6531923237839907477926238557)
1071
(2.700000000000000000000000000,0.6581365913145316881956101946)
1072
(2.720000000000000000000000000,0.6630131974925924426897085579)
1073
(2.740000000000000000000000000,0.6678228573551567374966233890)
1074
(2.760000000000000000000000000,0.6725662903900527205969351390)
1075
(2.780000000000000000000000000,0.6772442197671033774909918137)
1076
(2.800000000000000000000000000,0.6818573716092565606803011878)
1077
(2.820000000000000000000000000,0.6864064743021509464380001181)
1078
(2.840000000000000000000000000,0.6908922578406164741587417472)
1079
(2.860000000000000000000000000,0.6953154532106499224666013469)
1080
(2.880000000000000000000000000,0.6996767918054481450854513948)
1081
(2.900000000000000000000000000,0.7039770048741230491391654640)
1082
(2.920000000000000000000000000,0.7082168230017635775685842064)
1083
(2.940000000000000000000000000,0.7123969756195506913841508629)
1084
(2.960000000000000000000000000,0.7165181905436715788488349306)
1085
(2.980000000000000000000000000,0.7205811935418189959745336284)
1086
(3.000000000000000000000000000,0.7245867079261007203076142097)
1087
1088
\pscircle[linecolor=red](1,0){0.1}
1089
\end{pspicture}
1090
\caption{Curves of rank 1}\label{curvesof1}
1091
\end{center}
1092
\end{figure}
1093
1094
We can also compare the $L$-series of various curves all of rank 2 (Figure~\ref{curvesof2}).
1095
\begin{figure}
1096
\begin{center}
1097
\psset{unit=.9in}
1098
\begin{pspicture}(-0.5,-1.5)(3,1.5)
1099
\psgrid[gridcolor=gray]
1100
1101
% axes
1102
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
1103
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
1104
1105
\psline[linecolor=blue]
1106
(0.01999999999999999999999999999,0.06633205232924863831021756275)
1107
(0.03999999999999999999999999999,0.1238589869993040262250816535)
1108
(0.05999999999999999999999999999,0.1732490974204031901524233845)
1109
(0.07999999999999999999999999999,0.2151452257836850350849844517)
1110
(0.09999999999999999999999999999,0.2501634032128948577781162685)
1111
(0.1199999999999999999999999999,0.2788918829395867975266200164)
1112
(0.1399999999999999999999999999,0.3018905175888066893166510691)
1113
(0.1599999999999999999999999999,0.3196904360975578975797463927)
1114
(0.1799999999999999999999999999,0.3327939799261070683793316003)
1115
(0.1999999999999999999999999999,0.3416748620723204017123107521)
1116
(0.2199999999999999999999999999,0.3467785159727270810670498189)
1117
(0.2399999999999999999999999999,0.3485226046826493404029088088)
1118
(0.2599999999999999999999999999,0.3472976637838162485768392706)
1119
(0.2799999999999999999999999999,0.3434678542840188216894816108)
1120
(0.2999999999999999999999999999,0.3373718043623533659424285446)
1121
(0.3199999999999999999999999999,0.3293235211882361546026495165)
1122
(0.3399999999999999999999999999,0.3196133562153448294365761606)
1123
(0.3599999999999999999999999999,0.3085090093354303941233055163)
1124
(0.3799999999999999999999999999,0.2962565590837378578062775966)
1125
(0.3999999999999999999999999999,0.2830815077294071810755392342)
1126
(0.4199999999999999999999999999,0.2691898315721259382198436763)
1127
(0.4399999999999999999999999999,0.2547690281114520602682614897)
1128
(0.4599999999999999999999999999,0.2399891529681262198170412718)
1129
(0.4799999999999999999999999999,0.2250038405273628209988407423)
1130
(0.5000000000000000000000000000,0.2099513032520556529211768679)
1131
(0.5199999999999999999999999999,0.1949553054880592575852744770)
1132
(0.5399999999999999999999999999,0.1801261083627015878193116810)
1133
(0.5599999999999999999999999999,0.1655613830694359537456005338)
1134
(0.5799999999999999999999999999,0.1513470904435447588671322255)
1135
(0.5999999999999999999999999999,0.1375583252730762514803768786)
1136
(0.6199999999999999999999999999,0.1242601242622731563050750119)
1137
(0.6400000000000000000000000000,0.1115082369777323362159891224)
1138
(0.6600000000000000000000000000,0.09934985946607825450425559998)
1139
(0.6800000000000000000000000000,0.08782433054128655920915869191)
1140
(0.6999999999999999999999999999,0.07696379100480972080761890705)
1141
(0.7200000000000000000000000000,0.06679380628681199812990246179)
1142
(0.7400000000000000000000000000,0.05733395318623945854793812367)
1143
(0.7600000000000000000000000000,0.04859837154492148960574052244)
1144
(0.7800000000000000000000000000,0.04059628181989797667737901674)
1145
(0.8000000000000000000000000000,0.03333246962187091404798015893)
1146
(0.8200000000000000000000000000,0.02680773836899450420488153651)
1147
(0.8400000000000000000000000000,0.02101933126678998630035469414)
1148
(0.8600000000000000000000000000,0.01596132386920678110544201227)
1149
(0.8800000000000000000000000000,0.01162498850493334187192106899)
1150
(0.9000000000000000000000000000,0.007999131868965912642441395577)
1151
(0.9200000000000000000000000000,0.005070407083956478888872728231)
1152
(0.9400000000000000000000000000,0.002823601530591176909816049109)
1153
(0.9600000000000000000000000000,0.001241901732643167124218419275)
1154
(0.9800000000000000000000000000,0.0003071365616956409405041744633)
1155
(1.000000000000000000000000000,0)
1156
(1.020000000000000000000000000,0.0003002546685542481584883817277)
1157
(1.040000000000000000000000000,0.001186917292175145886470395270)
1158
(1.060000000000000000000000000,0.002638427234907597995192163478)
1159
(1.080000000000000000000000000,0.004632799198283402720258234366)
1160
(1.100000000000000000000000000,0.007147761132339355951297819566)
1161
(1.120000000000000000000000000,0.01016087836548349682017904803)
1162
(1.140000000000000000000000000,0.01364966491473845803352122823)
1163
(1.160000000000000000000000000,0.01759168289300856828113180647)
1164
(1.180000000000000000000000000,0.02196463088517071480962718469)
1165
(1.200000000000000000000000000,0.02674642212028410492580900504)
1166
(1.220000000000000000000000000,0.03191525322331037831235999896)
1167
(1.240000000000000000000000000,0.03744966428665069684926975382)
1168
(1.260000000000000000000000000,0.04332859095972094819532206158)
1169
(1.280000000000000000000000000,0.04953140921384744528365415091)
1170
(1.300000000000000000000000000,0.05603797340009158542754001716)
1171
(1.320000000000000000000000000,0.06282864817929507627563652885)
1172
(1.340000000000000000000000000,0.06988433486674700883909261771)
1173
(1.360000000000000000000000000,0.07718649269845979080186917044)
1174
(1.380000000000000000000000000,0.08471715549213489220344650360)
1175
(1.400000000000000000000000000,0.09245894414351856480180224416)
1176
(1.420000000000000000000000000,0.1003950753679962330352313589)
1177
(1.440000000000000000000000000,0.1085093670679450315037515450)
1178
(1.460000000000000000000000000,0.1167862406785404202910496723)
1179
(1.480000000000000000000000000,0.1252107208183704033015454849)
1180
(1.500000000000000000000000000,0.1337684325463184144182903735)
1181
(1.520000000000000000000000000,0.1424455965026967638450474248)
1182
(1.540000000000000000000000000,0.1512290221905055775109462058)
1183
(1.560000000000000000000000000,0.1601060996319128505769381796)
1184
(1.580000000000000000000000000,0.1690647896155523320460112270)
1185
(1.600000000000000000000000000,0.1780936127319682621754391509)
1186
(1.620000000000000000000000000,0.1871816373774489639875751267)
1187
(1.640000000000000000000000000,0.1963184668905336185903967618)
1188
(1.660000000000000000000000000,0.2054942259705965816493012890)
1189
(1.680000000000000000000000000,0.2146995465140597545581439636)
1190
(1.700000000000000000000000000,0.2239255529909046600601038464)
1191
(1.720000000000000000000000000,0.2331638474722015517620026003)
1192
(1.740000000000000000000000000,0.2424064944082936284211799523)
1193
(1.760000000000000000000000000,0.2516460052470219006656846127)
1194
(1.780000000000000000000000000,0.2608753229719034621687023556)
1195
(1.800000000000000000000000000,0.2700878066314372903924979781)
1196
(1.820000000000000000000000000,0.2792772159226632351689109758)
1197
(1.840000000000000000000000000,0.2884376958846991638357231880)
1198
(1.860000000000000000000000000,0.2975637617511876160398143306)
1199
(1.880000000000000000000000000,0.3066502840043577897819130821)
1200
(1.900000000000000000000000000,0.3156924736677139737711561023)
1201
(1.920000000000000000000000000,0.3246858678691621343664513934)
1202
(1.940000000000000000000000000,0.3336263157016484572980805859)
1203
(1.960000000000000000000000000,0.3425099644040751384131523910)
1204
(1.980000000000000000000000000,0.3513332458813491930824106960)
1205
(2.000000000000000000000000000,0.3600928635788807296980040920)
1206
(2.020000000000000000000000000,0.3687857797236508258718531647)
1207
(2.040000000000000000000000000,0.3774092029410902247245086278)
1208
(2.060000000000000000000000000,0.3859605762544244019818914223)
1209
(2.080000000000000000000000000,0.3944375654708254587933895777)
1210
(2.100000000000000000000000000,0.4028380479566454785017768724)
1211
(2.120000000000000000000000000,0.4111601018021694913778216111)
1212
(2.140000000000000000000000000,0.4194019953747003403784799398)
1213
(2.160000000000000000000000000,0.4275621772573550736243327165)
1214
(2.180000000000000000000000000,0.4356392665696967076039051483)
1215
(2.200000000000000000000000000,0.4436320436652311156968651002)
1216
(2.220000000000000000000000000,0.4515394411998522541716632602)
1217
(2.240000000000000000000000000,0.4593605355645067898736147499)
1218
(2.260000000000000000000000000,0.4670945386746592267266241759)
1219
(2.280000000000000000000000000,0.4747407901085595146246071222)
1220
(2.300000000000000000000000000,0.4822987495858363724860086952)
1221
(2.320000000000000000000000000,0.4897679897775514617778094699)
1222
(2.340000000000000000000000000,0.4971481894385431413804738794)
1223
(2.360000000000000000000000000,0.5044391268526555466528667942)
1224
(2.380000000000000000000000000,0.5116406735812815425491346379)
1225
(2.400000000000000000000000000,0.5187527885055396888373777886)
1226
(2.420000000000000000000000000,0.5257755121523492800387175090)
1227
(2.440000000000000000000000000,0.5327089612946578700408722419)
1228
(2.460000000000000000000000000,0.5395533238161070432200454270)
1229
(2.480000000000000000000000000,0.5463088538304895934951863330)
1230
(2.500000000000000000000000000,0.5529758670464501924162602402)
1231
(2.520000000000000000000000000,0.5595547363680079383280405544)
1232
(2.540000000000000000000000000,0.5660458877216291221281075280)
1233
(2.560000000000000000000000000,0.5724497961007487054862753885)
1234
(2.580000000000000000000000000,0.5787669818188262865625489756)
1235
(2.600000000000000000000000000,0.5849980069622239219107713778)
1236
(2.620000000000000000000000000,0.5911434720344065483191188197)
1237
(2.640000000000000000000000000,0.5972040127831886220005883782)
1238
(2.660000000000000000000000000,0.6031802972029809125617581586)
1239
(2.680000000000000000000000000,0.6090730227042273154751206094)
1240
(2.700000000000000000000000000,0.6148829134424614342827547772)
1241
(2.720000000000000000000000000,0.6206107177996550663633346026)
1242
(2.740000000000000000000000000,0.6262572060107743017435208269)
1243
(2.760000000000000000000000000,0.6318231679287025612292137093)
1244
(2.780000000000000000000000000,0.6373094109209325434200891978)
1245
(2.800000000000000000000000000,0.6427167578916698305616933645)
1246
(2.820000000000000000000000000,0.6480460454232290454373140670)
1247
(2.840000000000000000000000000,0.6532981220308382842075416019)
1248
(2.860000000000000000000000000,0.6584738465251984961861606039)
1249
(2.880000000000000000000000000,0.6635740864773710494218020169)
1250
(2.900000000000000000000000000,0.6685997167807884964349752979)
1251
(2.920000000000000000000000000,0.6735516183054001932121618872)
1252
(2.940000000000000000000000000,0.6784306766391756452179075042)
1253
(2.960000000000000000000000000,0.6832377809123940319920346023)
1254
(2.980000000000000000000000000,0.6879738227003481228525514787)
1255
(3.000000000000000000000000000,0.6926396950002846117222293821)
1256
1257
\psline[linecolor=green]
1258
(0.01999999999999999999999999999,0.08659984945133674338906694595)
1259
(0.03999999999999999999999999999,0.1615661035703338399800762145)
1260
(0.05999999999999999999999999999,0.2257993055327098452727697659)
1261
(0.07999999999999999999999999999,0.2801637906136125036053488751)
1262
(0.09999999999999999999999999999,0.3254860829631143831922152189)
1263
(0.1199999999999999999999999999,0.3625538088376822623864758615)
1264
(0.1399999999999999999999999999,0.3921150599280513245285173171)
1265
(0.1599999999999999999999999999,0.4148781466340938403697318837)
1266
(0.1799999999999999999999999999,0.4315116869092438138304586224)
1267
(0.1999999999999999999999999999,0.4426449816483569435653430304)
1268
(0.2199999999999999999999999999,0.4488686325421619730084076064)
1269
(0.2399999999999999999999999999,0.4507353628880282299994366560)
1270
(0.2599999999999999999999999999,0.4487610060503827941140766953)
1271
(0.2799999999999999999999999999,0.4434256301246638820000691469)
1272
(0.2999999999999999999999999999,0.4351747708960400513387784739)
1273
(0.3199999999999999999999999999,0.4244207484178648212894442708)
1274
(0.3399999999999999999999999999,0.4115440454841957291521629486)
1275
(0.3599999999999999999999999999,0.3968947289544053432924392484)
1276
(0.3799999999999999999999999999,0.3807938973240728209823875697)
1277
(0.3999999999999999999999999999,0.3635351401424236080327924600)
1278
(0.4199999999999999999999999999,0.3453859968693156113445656312)
1279
(0.4399999999999999999999999999,0.3265894045601272422603527880)
1280
(0.4599999999999999999999999999,0.3073651253800775043005893098)
1281
(0.4799999999999999999999999999,0.2879111463948963630459325364)
1282
(0.5000000000000000000000000000,0.2684050453759618648351203860)
1283
(0.5199999999999999999999999999,0.2490053175078321856910200210)
1284
(0.5399999999999999999999999999,0.2298526589065475500756402011)
1285
(0.5599999999999999999999999999,0.2110712037594157547545607728)
1286
(0.5799999999999999999999999999,0.1927697126917398274646895829)
1287
(0.5999999999999999999999999999,0.1750427106628929971098958984)
1288
(0.6199999999999999999999999999,0.1579715733023999087412417719)
1289
(0.6400000000000000000000000000,0.1416255611246879406556313839)
1290
(0.6600000000000000000000000000,0.1260628015167421208266439057)
1291
(0.6800000000000000000000000000,0.1113312187832460059972981905)
1292
(0.6999999999999999999999999999,0.09746941286556627726510682110)
1293
(0.7200000000000000000000000000,0.08450748763025262593826273864)
1294
(0.7400000000000000000000000000,0.07246782985518477880483892322)
1295
(0.7600000000000000000000000000,0.06136584023224035304210141152)
1296
(0.7800000000000000000000000000,0.05121061785907304676658181181)
1297
(0.8000000000000000000000000000,0.04200559981355947665645941811)
1298
(0.8200000000000000000000000000,0.03374915749658866795509447131)
1299
(0.8400000000000000000000000000,0.02643515149566672329932630884)
1300
(0.8600000000000000000000000000,0.02005344676649449270532118579)
1301
(0.8800000000000000000000000000,0.01459038995514483141812665236)
1302
(0.9000000000000000000000000000,0.01002925069233118696540467867)
1303
(0.9200000000000000000000000000,0.006350628685872168913872533391)
1304
(0.9400000000000000000000000000,0.003532828419928195022977245567)
1305
(0.9600000000000000000000000000,0.001552203241806163315448113929)
1306
(0.9800000000000000000000000000,0.0003834705807838703376444870406)
1307
(1.000000000000000000000000000,0)
1308
(1.020000000000000000000000000,0.0003740757333214403785942887469)
1309
(1.040000000000000000000000000,0.001477135305417070849202523675)
1310
(1.060000000000000000000000000,0.003279985776075738159476605629)
1311
(1.080000000000000000000000000,0.005752999090018736966692790321)
1312
(1.100000000000000000000000000,0.008866287951871656343229144468)
1313
(1.120000000000000000000000000,0.01258986358327106922924655184)
1314
(1.140000000000000000000000000,0.01689377665588876686834884923)
1315
(1.160000000000000000000000000,0.02174824263097644611619763557)
1316
(1.180000000000000000000000000,0.02712375267330779085349512976)
1317
(1.200000000000000000000000000,0.03299117124549599661978539455)
1318
(1.220000000000000000000000000,0.03932182142790586931262303508)
1319
(1.240000000000000000000000000,0.04608755895001985962565277415)
1320
(1.260000000000000000000000000,0.05326083586136826475222989807)
1321
(1.280000000000000000000000000,0.06081475471416463737756436323)
1322
(1.300000000000000000000000000,0.06872311407573028922553065128)
1323
(1.320000000000000000000000000,0.07696044613674608704050478108)
1324
(1.340000000000000000000000000,0.08550204713140635482517414422)
1325
(1.360000000000000000000000000,0.09432400123771457185640588926)
1326
(1.380000000000000000000000000,0.1034031985804780874820206116)
1327
(1.400000000000000000000000000,0.1127173479160350913420693261)
1328
(1.420000000000000000000000000,0.1222449845363715304978419280)
1329
(1.440000000000000000000000000,0.1319654738910350225500573357)
1330
(1.460000000000000000000000000,0.1418590113880922099939280137)
1331
(1.480000000000000000000000000,0.1519066188002611420751043017)
1332
(1.500000000000000000000000000,0.1620901376692291287744681185)
1333
(1.520000000000000000000000000,0.1723922200699808247371605793)
1334
(1.540000000000000000000000000,0.1827963170676478932587558463)
1335
(1.560000000000000000000000000,0.1932866651718835445269562706)
1336
(1.580000000000000000000000000,0.2038482710679928759694941426)
1337
(1.600000000000000000000000000,0.2144668948799417546596904049)
1338
(1.620000000000000000000000000,0.2251290321978503805742760655)
1339
(1.640000000000000000000000000,0.2358218950815796453505119813)
1340
(1.660000000000000000000000000,0.2465333922324660940423400183)
1341
(1.680000000000000000000000000,0.2572521085070824798886451237)
1342
(1.700000000000000000000000000,0.2679672839300243868821664081)
1343
(1.720000000000000000000000000,0.2786687923470793744286170515)
1344
(1.740000000000000000000000000,0.2893471198456554250798688435)
1345
(1.760000000000000000000000000,0.2999933430559638899255570578)
1346
(1.780000000000000000000000000,0.3105991074341044342672673737)
1347
(1.800000000000000000000000000,0.3211566056168237035839211179)
1348
(1.820000000000000000000000000,0.3316585559272558824306471086)
1349
(1.840000000000000000000000000,0.3420981811013447159985118908)
1350
(1.860000000000000000000000000,0.3524691872958380432049921184)
1351
(1.880000000000000000000000000,0.3627657434306850390236320890)
1352
(1.900000000000000000000000000,0.3729824609113032214475972544)
1353
(1.920000000000000000000000000,0.3831143737694693199894083870)
1354
(1.940000000000000000000000000,0.3931569192554802085364406101)
1355
(1.960000000000000000000000000,0.4031059189086845195519800008)
1356
(1.980000000000000000000000000,0.4129575601284618337169142686)
1357
(2.000000000000000000000000000,0.4227083782631862840531300173)
1358
(2.020000000000000000000000000,0.4323552392306190143077202567)
1359
(2.040000000000000000000000000,0.4418953226794952866039594729)
1360
(2.060000000000000000000000000,0.4513261056987752763617030270)
1361
(2.080000000000000000000000000,0.4606453470780828122001156439)
1362
(2.100000000000000000000000000,0.4698510721202354788400152059)
1363
(2.120000000000000000000000000,0.4789415580044463590353816931)
1364
(2.140000000000000000000000000,0.4879153196967277146590522256)
1365
(2.160000000000000000000000000,0.4967710964022271952498960097)
1366
(2.180000000000000000000000000,0.5055078385526563614481417012)
1367
(2.200000000000000000000000000,0.5141246953206095370002406974)
1368
(2.220000000000000000000000000,0.5226210026513997640926836303)
1369
(2.240000000000000000000000000,0.5309962718020407559288804593)
1370
(2.260000000000000000000000000,0.5392501783761632827585564141)
1371
(2.280000000000000000000000000,0.5473825518429566284805219020)
1372
(2.300000000000000000000000000,0.5553933655276569514519371321)
1373
(2.320000000000000000000000000,0.5632827270606519474428864747)
1374
(2.340000000000000000000000000,0.5710508692719234885432838264)
1375
(2.360000000000000000000000000,0.5786981415172961536854436108)
1376
(2.380000000000000000000000000,0.5862250014227898811795853347)
1377
(2.400000000000000000000000000,0.5936320070332802651541766653)
1378
(2.420000000000000000000000000,0.6009198093516419341537427330)
1379
(2.440000000000000000000000000,0.6080891452545813335907100271)
1380
(2.460000000000000000000000000,0.6151408307714480731030904487)
1381
(2.480000000000000000000000000,0.6220757547124423856229418897)
1382
(2.500000000000000000000000000,0.6288948726328043267828583709)
1383
(2.520000000000000000000000000,0.6355992011197727899635821041)
1384
(2.540000000000000000000000000,0.6421898123893343739615311661)
1385
(2.560000000000000000000000000,0.6486678291800392129231469344)
1386
(2.580000000000000000000000000,0.6550344199314390703236944569)
1387
(2.600000000000000000000000000,0.6612907942349986999984414566)
1388
(2.620000000000000000000000000,0.6674381985456414290239729303)
1389
(2.640000000000000000000000000,0.6734779121424111854128655007)
1390
(2.660000000000000000000000000,0.6794112433270631426183347692)
1391
(2.680000000000000000000000000,0.6852395258497314219336064908)
1392
(2.700000000000000000000000000,0.6909641155511627745245384405)
1393
(2.720000000000000000000000000,0.6965863872113479800439067208)
1394
(2.740000000000000000000000000,0.7021077315947261836104004835)
1395
(2.760000000000000000000000000,0.7075295526824800765027069406)
1396
(2.780000000000000000000000000,0.7128532650827804146123687987)
1397
(2.800000000000000000000000000,0.7180802916101757306526349860)
1398
(2.820000000000000000000000000,0.7232120610256562468029833171)
1399
(2.840000000000000000000000000,0.7282500059292490833453226164)
1400
(2.860000000000000000000000000,0.7331955607973241570434771109)
1401
(2.880000000000000000000000000,0.7380501601571060519215471069)
1402
(2.900000000000000000000000000,0.7428152368911961058441952577)
1403
(2.920000000000000000000000000,0.7474922206652105600776933923)
1404
(2.940000000000000000000000000,0.7520825364719345181204989493)
1405
(2.960000000000000000000000000,0.7565876032856773797499068849)
1406
(2.980000000000000000000000000,0.7610088328207931470124507230)
1407
(3.000000000000000000000000000,0.7653476283885983898024055268)
1408
1409
\psline[linecolor=cyan]
1410
(0.01999999999999999999999999999,0.1687347390041907194271013526)
1411
(0.03999999999999999999999999999,0.3097634558764556878598930488)
1412
(0.05999999999999999999999999999,0.4260250289078247711405460089)
1413
(0.07999999999999999999999999999,0.5202336384427707305577068782)
1414
(0.09999999999999999999999999999,0.5948882835605865511463336347)
1415
(0.1199999999999999999999999999,0.6522829399719607003565582162)
1416
(0.1399999999999999999999999999,0.6945171376650787938688203174)
1417
(0.1599999999999999999999999999,0.7235067715224408074513862224)
1418
(0.1799999999999999999999999999,0.7409949886703663095230821789)
1419
(0.1999999999999999999999999999,0.7485630231233999977790755376)
1420
(0.2199999999999999999999999999,0.7476408717201448770248118507)
1421
(0.2399999999999999999999999999,0.7395177257603995710779455697)
1422
(0.2599999999999999999999999999,0.7253520904625466886476138588)
1423
(0.2799999999999999999999999999,0.7061815396548235879044178900)
1424
(0.2999999999999999999999999999,0.6829320662590333355697303581)
1425
(0.3199999999999999999999999999,0.6564270003613377461101402900)
1426
(0.3399999999999999999999999999,0.6273954762107698045039245970)
1427
(0.3599999999999999999999999999,0.5964804375401305040648602079)
1428
(0.3799999999999999999999999999,0.5642461773449760188251718458)
1429
(0.3999999999999999999999999999,0.5311854138457399761598717927)
1430
(0.4199999999999999999999999999,0.4977259089406478416179562726)
1431
(0.4399999999999999999999999999,0.4642366391629550718109864887)
1432
(0.4599999999999999999999999999,0.4310335321014350951612283706)
1433
(0.4799999999999999999999999999,0.3983847835316671834501658460)
1434
(0.5000000000000000000000000000,0.3665157722298024291701906421)
1435
(0.5199999999999999999999999999,0.3356135906819979314135332868)
1436
(0.5399999999999999999999999999,0.3058312107340588520323246191)
1437
(0.5599999999999999999999999999,0.2772913037109434376070466300)
1438
(0.5799999999999999999999999999,0.2500897347309025947815121686)
1439
(0.5999999999999999999999999999,0.2242987508934534348594040060)
1440
(0.6199999999999999999999999999,0.1999698827772171758036086123)
1441
(0.6400000000000000000000000000,0.1771365782804071241977294154)
1442
(0.6600000000000000000000000000,0.1558165873059753671793353577)
1443
(0.6800000000000000000000000000,0.1360141151632230866638291331)
1444
(0.6999999999999999999999999999,0.1177217618522093956485387441)
1445
(0.7200000000000000000000000000,0.1009222636372187105264532685)
1446
(0.7400000000000000000000000000,0.08559005251843382977102813337)
1447
(0.7600000000000000000000000000,0.07169264839164885796431825433)
1448
(0.7800000000000000000000000000,0.05919189785678083540309754206)
1449
(0.8000000000000000000000000000,0.04804507280743565708318706483)
1450
(0.8200000000000000000000000000,0.03820584111434883356398280721)
1451
(0.8400000000000000000000000000,0.02962512091205089936514899244)
1452
(0.8600000000000000000000000000,0.02225182921610899200196565302)
1453
(0.8800000000000000000000000000,0.01603353484207978021979604229)
1454
(0.9000000000000000000000000000,0.01091702487015336040560023746)
1455
(0.9200000000000000000000000000,0.006848793203770159779441394323)
1456
(0.9400000000000000000000000000,0.003775459107900518256396867657)
1457
(0.9600000000000000000000000000,0.001644122984202839891346984812)
1458
(0.9800000000000000000000000000,0.0004026660464025739443104774379)
1459
(1.000000000000000000000000000,0)
1460
(1.020000000000000000000000000,0.0003862723054948066617406952477)
1461
(1.040000000000000000000000000,0.001513032113088499876421141767)
1462
(1.060000000000000000000000000,0.003333361498433251023685428347)
1463
(1.080000000000000000000000000,0.005801976202393113650108972135)
1464
(1.100000000000000000000000000,0.008875299681783359953704217982)
1465
(1.120000000000000000000000000,0.01251151391136381409512168426)
1466
(1.140000000000000000000000000,0.01667059003861724546596810549)
1467
(1.160000000000000000000000000,0.02131430168063283622949173513)
1468
(1.180000000000000000000000000,0.02640622336530277482890138278)
1469
(1.200000000000000000000000000,0.03191171635558796532676618116)
1470
(1.220000000000000000000000000,0.03779790385438855690814664270)
1471
(1.240000000000000000000000000,0.04403363736716211103991519355)
1472
(1.260000000000000000000000000,0.05058945579849629168173536722)
1473
(1.280000000000000000000000000,0.05743753867603550401473552744)
1474
(1.300000000000000000000000000,0.06455165472920156868719088127)
1475
(1.320000000000000000000000000,0.07190710689980966332184807688)
1476
(1.340000000000000000000000000,0.07948067472579069194707263335)
1477
(1.360000000000000000000000000,0.08725055491667627718858504307)
1478
(1.380000000000000000000000000,0.09519630082922816849409064701)
1479
(1.400000000000000000000000000,0.1032987614526049340257431521)
1480
(1.420000000000000000000000000,0.1115400204238193699565477873)
1481
(1.440000000000000000000000000,0.1199033355150723486086693114)
1482
(1.460000000000000000000000000,0.1283730789640319539595528609)
1483
(1.480000000000000000000000000,0.1369346789554950542593816227)
1484
(1.500000000000000000000000000,0.1455745625074095512152170283)
1485
(1.520000000000000000000000000,0.1542800999652881808773974369)
1486
(1.540000000000000000000000000,0.1630395512659965451612502705)
1487
(1.560000000000000000000000000,0.1718420140941831398058998512)
1488
(1.580000000000000000000000000,0.1806773740217156977614965211)
1489
(1.600000000000000000000000000,0.1895362566919159949566320288)
1490
(1.620000000000000000000000000,0.1984099820857033682684557335)
1491
(1.640000000000000000000000000,0.2072905208855613642470850309)
1492
(1.660000000000000000000000000,0.2161704529351624038425840220)
1493
(1.680000000000000000000000000,0.2250429277771845026664472354)
1494
(1.700000000000000000000000000,0.2339016272390242691226293313)
1495
(1.720000000000000000000000000,0.2427407300254717861948025714)
1496
(1.740000000000000000000000000,0.2515548782687115422795010329)
1497
(1.760000000000000000000000000,0.2603391459790191597235040295)
1498
(1.780000000000000000000000000,0.2690890093340281637276165956)
1499
(1.800000000000000000000000000,0.2778003187402566330964547885)
1500
(1.820000000000000000000000000,0.2864692725975411462099506779)
1501
(1.840000000000000000000000000,0.2950923926949729887540814649)
1502
(1.860000000000000000000000000,0.3036665011657328346224839520)
1503
(1.880000000000000000000000000,0.3121886989277531170473487551)
1504
(1.900000000000000000000000000,0.3206563455372932484041791275)
1505
(1.920000000000000000000000000,0.3290670403831948399176046587)
1506
(1.940000000000000000000000000,0.3374186051507060915754838194)
1507
(1.960000000000000000000000000,0.3457090674852504005013925226)
1508
(1.980000000000000000000000000,0.3539366457882967373412324986)
1509
(2.000000000000000000000000000,0.3620997350795093073749619684)
1510
(2.020000000000000000000000000,0.3701968938615595803086612145)
1511
(2.040000000000000000000000000,0.3782268319263296434771505013)
1512
(2.060000000000000000000000000,0.3861883990436826173173072790)
1513
(2.080000000000000000000000000,0.3940805744764894787184931210)
1514
(2.100000000000000000000000000,0.4019024572681527211151866557)
1515
(2.120000000000000000000000000,0.4096532572514307322332073675)
1516
(2.140000000000000000000000000,0.4173322867299212607343291828)
1517
(2.160000000000000000000000000,0.4249389527860898990057416090)
1518
(2.180000000000000000000000000,0.4324727501722151356162760776)
1519
(2.200000000000000000000000000,0.4399332547430528644075261555)
1520
(2.220000000000000000000000000,0.4473201173913902349882637008)
1521
(2.240000000000000000000000000,0.4546330584499533873154651578)
1522
(2.260000000000000000000000000,0.4618718625253497115287111653)
1523
(2.280000000000000000000000000,0.4690363737318581500317642190)
1524
(2.300000000000000000000000000,0.4761264912949274000608574941)
1525
(2.320000000000000000000000000,0.4831421654961995373351537021)
1526
(2.340000000000000000000000000,0.4900833939337444239404353274)
1527
(2.360000000000000000000000000,0.4969502180729680023491294069)
1528
(2.380000000000000000000000000,0.5037427200653456549603293514)
1529
(2.400000000000000000000000000,0.5104610198137312780793556475)
1530
(2.420000000000000000000000000,0.5171052722645051424875748460)
1531
(2.440000000000000000000000000,0.5236756649082509691463389921)
1532
(2.460000000000000000000000000,0.5301724154719972557737619718)
1533
(2.480000000000000000000000000,0.5365957697873223339549181815)
1534
(2.500000000000000000000000000,0.5429459998198097102067361150)
1535
(2.520000000000000000000000000,0.5492234018464528951116289384)
1536
(2.540000000000000000000000000,0.5554282947686502072054050471)
1537
(2.560000000000000000000000000,0.5615610185494030757589943169)
1538
(2.580000000000000000000000000,0.5676219327642393158478790169)
1539
(2.600000000000000000000000000,0.5736114152562288718831994464)
1540
(2.620000000000000000000000000,0.5795298608862467639870205535)
1541
(2.640000000000000000000000000,0.5853776803703695266500895599)
1542
(2.660000000000000000000000000,0.5911552991969703447973793718)
1543
(2.680000000000000000000000000,0.5968631566167073407973418926)
1544
(2.700000000000000000000000000,0.6025017046991819361513580966)
1545
(2.720000000000000000000000000,0.6080714074505827006797631584)
1546
(2.740000000000000000000000000,0.6135727399871273082238144510)
1547
(2.760000000000000000000000000,0.6190061877595737355488739677)
1548
(2.780000000000000000000000000,0.6243722458244941572078421977)
1549
(2.800000000000000000000000000,0.6296714181583934810149355012)
1550
(2.820000000000000000000000000,0.6349042170111114033781276632)
1551
(2.840000000000000000000000000,0.6400711622952743974146266533)
1552
(2.860000000000000000000000000,0.6451727810088642262107260651)
1553
(2.880000000000000000000000000,0.6502096066882443373084094766)
1554
(2.900000000000000000000000000,0.6551821788892366749732924599)
1555
(2.920000000000000000000000000,0.6600910426940707730076970068)
1556
(2.940000000000000000000000000,0.6649367482422360912766047459)
1557
(2.960000000000000000000000000,0.6697198502834589648931187917)
1558
(2.980000000000000000000000000,0.6744409077511986835438382299)
1559
(3.000000000000000000000000000,0.6791004833552144569456907065)
1560
1561
\psline[linecolor=magenta]
1562
(0.01999999999999999999999999999,0.7333974445416658508817187540)
1563
(0.03999999999999999999999999999,1.322593364693086437073555786)
1564
(0.05999999999999999999999999999,1.787029925133150417352153303)
1565
(0.07999999999999999999999999999,2.144053816101128793326022417)
1566
(0.09999999999999999999999999999,2.409094935216228171682073272)
1567
(0.1199999999999999999999999999,2.595835687426499240982864389)
1568
(0.1399999999999999999999999999,2.716370416234184779647485251)
1569
(0.1599999999999999999999999999,2.781354725057489550106635032)
1570
(0.1799999999999999999999999999,2.800144644874463566556642881)
1571
(0.1999999999999999999999999999,2.780925760531284677720411574)
1572
(0.2199999999999999999999999999,2.730832529700532998783748836)
1573
(0.2399999999999999999999999999,2.656058121007886414886786942)
1574
(0.2599999999999999999999999999,2.561955166116684300661159737)
1575
(0.2799999999999999999999999999,2.453127868712628542589449837)
1576
(0.2999999999999999999999999999,2.333515944924246675815913879)
1577
(0.3199999999999999999999999999,2.206470887794294404267418645)
1578
(0.3399999999999999999999999999,2.074825055579814231490172359)
1579
(0.3599999999999999999999999999,1.940954082109225277552471535)
1580
(0.3799999999999999999999999999,1.806833099027926473892211770)
1581
(0.3999999999999999999999999999,1.674087246088065354527083092)
1582
(0.4199999999999999999999999999,1.544036927996123653336923435)
1583
(0.4399999999999999999999999999,1.417738255815879896866202089)
1584
(0.4599999999999999999999999999,1.296019088436458970778935269)
1585
(0.4799999999999999999999999999,1.179511065895295663166962526)
1586
(0.5000000000000000000000000000,1.068678001994805356202066593)
1587
(0.5199999999999999999999999999,0.9638409791521500347795837548)
1588
(0.5399999999999999999999999999,0.8652004641564491336956499882)
1589
(0.5599999999999999999999999999,0.7728557397754898207668943471)
1590
(0.5799999999999999999999999999,0.6868219241820362384923548780)
1591
(0.5999999999999999999999999999,0.6070448281267605352034571042)
1592
(0.6199999999999999999999999999,0.5334138787901921831893435792)
1593
(0.6400000000000000000000000000,0.4657733193791694270861095844)
1594
(0.6600000000000000000000000000,0.4039318748404354129496643862)
1595
(0.6800000000000000000000000000,0.3476710565650106210761447551)
1596
(0.6999999999999999999999999999,0.2967522626503050804791535432)
1597
(0.7200000000000000000000000000,0.2509228151543967077819773810)
1598
(0.7400000000000000000000000000,0.2099210617873634506100612308)
1599
(0.7600000000000000000000000000,0.1734806565971455591923083118)
1600
(0.7800000000000000000000000000,0.1413341223741673000542254087)
1601
(0.8000000000000000000000000000,0.1132157866669970450917018702)
1602
(0.8200000000000000000000000000,0.08886417341469181742305270709)
1603
(0.8400000000000000000000000000,0.06802392320252407389682479984)
1604
(0.8600000000000000000000000000,0.05044730697840990872002654905)
1605
(0.8800000000000000000000000000,0.03589539066988258025867752831)
1606
(0.9000000000000000000000000000,0.02413890145939195531659444292)
1607
(0.9200000000000000000000000000,0.01495884045429858633070554571)
1608
(0.9400000000000000000000000000,0.008146881074554133551749220635)
1609
(0.9600000000000000000000000000,0.003505587625540175523364737809)
1610
(0.9800000000000000000000000000,0.0008484841783343084440206026595)
1611
(1.000000000000000000000000000,0)
1612
(1.020000000000000000000000000,0.0007953143203780905975066671107)
1613
(1.040000000000000000000000000,0.003080120150127838590093525975)
1614
(1.060000000000000000000000000,0.006710324141032327063677735431)
1615
(1.080000000000000000000000000,0.01155169707015954908740561578)
1616
(1.100000000000000000000000000,0.01747948740328051473997787455)
1617
(1.120000000000000000000000000,0.02437800852141411625451738712)
1618
(1.140000000000000000000000000,0.03214020855129847367761618335)
1619
(1.160000000000000000000000000,0.04066723030203137598478276298)
1620
(1.180000000000000000000000000,0.04986796755424304294643197774)
1621
(1.200000000000000000000000000,0.05965862285510501328102363457)
1622
(1.220000000000000000000000000,0.06996227102422392539665681365)
1623
(1.240000000000000000000000000,0.08070843175571211487881036069)
1624
(1.260000000000000000000000000,0.09183265399574265306090403127)
1625
(1.280000000000000000000000000,0.1032761141694155699881461289)
1626
(1.300000000000000000000000000,0.1149852298138535546478553566)
1627
(1.320000000000000000000000000,0.1269112897354042468945415239)
1628
(1.340000000000000000000000000,0.1390101014380670829529976229)
1629
(1.360000000000000000000000000,0.1512416562592208120163411090)
1630
(1.380000000000000000000000000,0.1635698123897662458301210842)
1631
(1.400000000000000000000000000,0.1759619957421218627974540083)
1632
(1.420000000000000000000000000,0.1883889184550832092014091589)
1633
(1.440000000000000000000000000,0.2008243146840336788995833474)
1634
(1.460000000000000000000000000,0.2132446932136466318543045928)
1635
(1.480000000000000000000000000,0.2256291063438762608753308486)
1636
(1.500000000000000000000000000,0.2379589344350261974965595522)
1637
(1.520000000000000000000000000,0.2502176854507871563827093955)
1638
(1.540000000000000000000000000,0.2623908088065245366509922886)
1639
(1.560000000000000000000000000,0.2744655228113072021201994549)
1640
(1.580000000000000000000000000,0.2864306549840506519927449198)
1641
(1.600000000000000000000000000,0.2982764945248346686924801909)
1642
(1.620000000000000000000000000,0.3099946562303307062501067090)
1643
(1.640000000000000000000000000,0.3215779551559425787872756901)
1644
(1.660000000000000000000000000,0.3330202913455257377278840693)
1645
(1.680000000000000000000000000,0.3443165439713781490088821110)
1646
(1.700000000000000000000000000,0.3554624742517134834000553605)
1647
(1.720000000000000000000000000,0.3664546365392918106818515301)
1648
(1.740000000000000000000000000,0.3772902970026672410237361030)
1649
(1.760000000000000000000000000,0.3879673593500903952396748702)
1650
(1.780000000000000000000000000,0.3984842970750388848969962752)
1651
(1.800000000000000000000000000,0.4088400917312804195020138932)
1652
(1.820000000000000000000000000,0.4190341767740062805561053654)
1653
(1.840000000000000000000000000,0.4290663865316704080009478556)
1654
(1.860000000000000000000000000,0.4389369099005430299270949451)
1655
(1.880000000000000000000000000,0.4486462483804914454106990177)
1656
(1.900000000000000000000000000,0.4581951780960238557970669403)
1657
(1.920000000000000000000000000,0.4675847154710949951688798533)
1658
(1.940000000000000000000000000,0.4768160862495202885868838185)
1659
(1.960000000000000000000000000,0.4858906975750453570819961310)
1660
(1.980000000000000000000000000,0.4948101128661547095180568582)
1661
(2.000000000000000000000000000,0.5035760292405769300637854358)
1662
(2.020000000000000000000000000,0.5121902572631650962020900537)
1663
(2.040000000000000000000000000,0.5206547028084217188189600817)
1664
(2.060000000000000000000000000,0.5289713508454260178785930181)
1665
(2.080000000000000000000000000,0.5371422509683426207894711075)
1666
(2.100000000000000000000000000,0.5451695045100840770601482617)
1667
(2.120000000000000000000000000,0.5530552530901074646736508357)
1668
(2.140000000000000000000000000,0.5608016684597925849202447921)
1669
(2.160000000000000000000000000,0.5684109435204219188402389654)
1670
(2.180000000000000000000000000,0.5758852843995073938205516874)
1671
(2.200000000000000000000000000,0.5832269034811328595227834978)
1672
(2.220000000000000000000000000,0.5904380132951503209346057096)
1673
(2.240000000000000000000000000,0.5975208211785279008955446270)
1674
(2.260000000000000000000000000,0.6044775246299425280822231190)
1675
(2.280000000000000000000000000,0.6113103072858833967377202215)
1676
(2.300000000000000000000000000,0.6180213354531246717423626266)
1677
(2.320000000000000000000000000,0.6246127551384773582881025787)
1678
(2.340000000000000000000000000,0.6310866895222785620414960232)
1679
(2.360000000000000000000000000,0.6374452368271575225750588705)
1680
(2.380000000000000000000000000,0.6436904685382659206784506742)
1681
(2.400000000000000000000000000,0.6498244279354072696288190006)
1682
(2.420000000000000000000000000,0.6558491289013770706074117842)
1683
(2.440000000000000000000000000,0.6617665549743603845326212084)
1684
(2.460000000000000000000000000,0.6675786586154533068080692035)
1685
(2.480000000000000000000000000,0.6732873606653045630406363746)
1686
(2.500000000000000000000000000,0.6788945499665364463340977492)
1687
(2.520000000000000000000000000,0.6844020831310223712129229683)
1688
(2.540000000000000000000000000,0.6898117844332916860593020191)
1689
(2.560000000000000000000000000,0.6951254458133198797275472853)
1690
(2.580000000000000000000000000,0.7003448269737613829109018246)
1691
(2.600000000000000000000000000,0.7054716555583089502042330240)
1692
(2.620000000000000000000000000,0.7105076273993330439047079281)
1693
(2.640000000000000000000000000,0.7154544068242805080495059273)
1694
(2.660000000000000000000000000,0.7203136270115068285167404039)
1695
(2.680000000000000000000000000,0.7250868903872921238673316856)
1696
(2.700000000000000000000000000,0.7297757690567584644771613146)
1697
(2.720000000000000000000000000,0.7343818052622750616659761371)
1698
(2.740000000000000000000000000,0.7389065118637173770242530532)
1699
(2.760000000000000000000000000,0.7433513728356445918291540461)
1700
(2.780000000000000000000000000,0.7477178437770847629719435788)
1701
(2.800000000000000000000000000,0.7520073524301753415738390579)
1702
(2.820000000000000000000000000,0.7562212992044049095062768312)
1703
(2.840000000000000000000000000,0.7603610577036458090907993685)
1704
(2.860000000000000000000000000,0.7644279752535621028686540966)
1705
(2.880000000000000000000000000,0.7684233734273278331819187999)
1706
(2.900000000000000000000000000,0.7723485485679012518941697869)
1707
(2.920000000000000000000000000,0.7762047723053755573206489098)
1708
(2.940000000000000000000000000,0.7799932920681693413414006019)
1709
(2.960000000000000000000000000,0.7837153315870337126617468554)
1710
(2.980000000000000000000000000,0.7873720913910409132404603198)
1711
(3.000000000000000000000000000,0.7909647492948838941071799245)
1712
1713
\rput(.55, 1.6){$E_4$}
1714
\rput(.35, .8){$E_3$}
1715
\rput(.3, .5){$E_2$}
1716
\rput(.4, .13){$E_1$}
1717
\rput(1.3,-1.55){$E_1=[0,1,1,-2,0],\ E_2=[1,0,0,0,1],\ E_3=[0,0,1,-199,1092],\ E_4=[1,0,0,-1,6]$}
1718
\pscircle[linecolor=red](1,0){0.1}
1719
\end{pspicture}
1720
\caption{Curves of rank 2}\label{curvesof2}
1721
\end{center}
1722
\end{figure}
1723
The conductors of these curves are 389, 433, 1001, and 3185 respectively. Note how the difference in conductors relates to the peak of the $L$-series between $x=0$ and $x=1$.
1724
1725
We make the similar comparison for some curves of rank 3 in Figure~\ref{curvesof3}. The three curves shown in this Figure are $E_1=[0,0,1,-7,6]$ which has conductor 5077, $E_2=[1,-1,1,-6,0]$ which has conductor 11197, and $E_3=[1,-1,0,-16,28]$ which has conductor 11642.
1726
\begin{figure}
1727
\begin{center}
1728
\psset{unit=.9in}
1729
\begin{pspicture}(-.5,-1.5)(3,1.5)
1730
\psgrid[gridcolor=gray]
1731
1732
% axes
1733
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
1734
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
1735
1736
\psline[linecolor=blue]
1737
(0.01999999999999999999999999999,-0.6941969565423001064579008505)
1738
(0.03999999999999999999999999999,-1.224303060054913888228611661)
1739
(0.05999999999999999999999999999,-1.617036990043072476380405432)
1740
(0.07999999999999999999999999999,-1.895599825260420064276982232)
1741
(0.09999999999999999999999999999,-2.080058567173062689443317490)
1742
(0.1199999999999999999999999999,-2.187697375097209442807822069)
1743
(0.1399999999999999999999999999,-2.233337785910732953117521202)
1744
(0.1599999999999999999999999999,-2.229629434932846597202591093)
1745
(0.1799999999999999999999999999,-2.187312956336142367783540870)
1746
(0.1999999999999999999999999999,-2.115456837327142818018141335)
1747
(0.2199999999999999999999999999,-2.021670043781688359424560791)
1748
(0.2399999999999999999999999999,-1.912292237330765763733304833)
1749
(0.2599999999999999999999999999,-1.792563374456563069666298951)
1750
(0.2799999999999999999999999999,-1.666774424759994937314205875)
1751
(0.2999999999999999999999999999,-1.538400874605764615403251437)
1752
(0.3199999999999999999999999999,-1.410220599075322086377357005)
1753
(0.3399999999999999999999999999,-1.284417593807022454714262376)
1754
(0.3599999999999999999999999999,-1.162672962285572963824604544)
1755
(0.3799999999999999999999999999,-1.046244456166366802310495314)
1756
(0.3999999999999999999999999999,-0.9360357684041337897681360899)
1757
(0.4199999999999999999999999999,-0.8326566829304489846775894104)
1758
(0.4399999999999999999999999999,-0.7364750916176941703400764989)
1759
(0.4599999999999999999999999999,-0.6476618001723166913083888297)
1760
(0.4799999999999999999999999999,-0.5662289600405636110221516461)
1761
(0.5000000000000000000000000000,-0.4920628837884663767162722526)
1762
(0.5199999999999999999999999999,-0.4249519269617209726671399220)
1763
(0.5399999999999999999999999999,-0.3646100502276295382286701887)
1764
(0.5599999999999999999999999999,-0.3106966116287169800938014682)
1765
(0.5799999999999999999999999999,-0.2628328799305563687087769934)
1766
(0.5999999999999999999999999999,-0.2206157061565192332634880189)
1767
(0.6199999999999999999999999999,-0.1836287412558685910948819996)
1768
(0.6400000000000000000000000000,-0.1514515432036038941355468859)
1769
(0.6600000000000000000000000000,-0.1236668764153878637362459807)
1770
(0.6800000000000000000000000000,-0.09986646990233779653602125192)
1771
(0.6999999999999999999999999999,-0.07965546780820809922285492823)
1772
(0.7200000000000000000000000000,-0.06265577658705412018384921392)
1773
(0.7400000000000000000000000000,-0.04850848682044001416921755886)
1774
(0.7600000000000000000000000000,-0.03687552427649674769076936318)
1775
(0.7800000000000000000000000000,-0.02744066402710006863491442098)
1776
(0.8000000000000000000000000000,-0.01991002302575148987446085203)
1777
(0.8200000000000000000000000000,-0.01401213028322172326308234058)
1778
(0.8400000000000000000000000000,-0.009497659451189876915738041398)
1779
(0.8600000000000000000000000000,-0.006138896041403990484653390642)
1780
(0.8800000000000000000000000000,-0.003729000489484266274466262582)
1781
(0.9000000000000000000000000000,-0.002081118652996614152056766787)
1782
(0.9200000000000000000000000000,-0.001027382961363314017406494273)
1783
(0.9400000000000000000000000000,-0.0004178401723634765122120044597)
1784
(0.9600000000000000000000000000,-0.0001193354108090775330798550141)
1785
(0.9800000000000000000000000000,-0.00001437675569523539982970834126)
1786
(1.000000000000000000000000000,0)
1787
(1.020000000000000000000000000,0.00001335076010323709396208854362)
1788
(1.040000000000000000000000000,0.0001029144255272643359860796470)
1789
(1.060000000000000000000000000,0.0003346666611260519448020428584)
1790
(1.080000000000000000000000000,0.0007643291131354103068160924266)
1791
(1.100000000000000000000000000,0.001438329075870453641136902059)
1792
(1.120000000000000000000000000,0.002394706281075908365750437174)
1793
(1.140000000000000000000000000,0.003663964864451018058842766187)
1794
(1.160000000000000000000000000,0.005269869692308771884970831179)
1795
(1.180000000000000000000000000,0.007230187143107076178798332141)
1796
(1.200000000000000000000000000,0.009557371165349786145716519883)
1797
(1.220000000000000000000000000,0.01225919600281414307728846704)
1798
(1.240000000000000000000000000,0.01533933741442756779897108478)
1799
(1.260000000000000000000000000,0.01879790454047723887660114808)
1800
(1.280000000000000000000000000,0.02263192479750214967619458858)
1801
(1.300000000000000000000000000,0.02683578433704244666524467576)
1802
(1.320000000000000000000000000,0.03140162669208463969482120891)
1803
(1.340000000000000000000000000,0.03631971227130751156193470885)
1804
(1.360000000000000000000000000,0.04157874135518094727701674422)
1805
(1.380000000000000000000000000,0.04716614320819906972809900905)
1806
(1.400000000000000000000000000,0.05306833385534747781079476116)
1807
(1.420000000000000000000000000,0.05927094498449120520526767588)
1808
(1.440000000000000000000000000,0.06575902633492268847102645806)
1809
(1.460000000000000000000000000,0.07251722382017324802586003591)
1810
(1.480000000000000000000000000,0.07952993551397631749826337090)
1811
(1.500000000000000000000000000,0.08678144750494967257462341313)
1812
(1.520000000000000000000000000,0.09425605150056515587276714319)
1813
(1.540000000000000000000000000,0.1019381459362585051211530930)
1814
(1.560000000000000000000000000,0.1098123222226609793674436171)
1815
(1.580000000000000000000000000,0.1178634376441322427302334033)
1816
(1.600000000000000000000000000,0.1260766763059780939693254887)
1817
(1.620000000000000000000000000,0.1344375994166443422785904058)
1818
(1.640000000000000000000000000,0.1429321860852856580248726532)
1819
(1.660000000000000000000000000,0.1515468657147449390448351677)
1820
(1.680000000000000000000000000,0.1602685429753365274249350349)
1821
(1.700000000000000000000000000,0.1690846162559851612628322842)
1822
(1.720000000000000000000000000,0.1779829904062208953656092390)
1823
(1.740000000000000000000000000,0.1869520845051851986279852277)
1824
(1.760000000000000000000000000,0.1959808353220252615207756937)
1825
(1.780000000000000000000000000,0.2050586970656590537713607681)
1826
(1.800000000000000000000000000,0.2141756379606673828509888437)
1827
(1.820000000000000000000000000,0.2233221341297726034172697453)
1828
(1.840000000000000000000000000,0.2324891612117428568921548218)
1829
(1.860000000000000000000000000,0.2416681840963529118286738355)
1830
(1.880000000000000000000000000,0.2508511451149711283894650525)
1831
(1.900000000000000000000000000,0.2600304509861604269868339229)
1832
(1.920000000000000000000000000,0.2691989587801167195171382618)
1833
(1.940000000000000000000000000,0.2783499611335646655554892373)
1834
(1.960000000000000000000000000,0.2874771709176397398850025740)
1835
(1.980000000000000000000000000,0.2965747055350691211609569031)
1836
(2.000000000000000000000000000,0.3056370709993943665549003166)
1837
(2.020000000000000000000000000,0.3146591459278403233288422136)
1838
(2.040000000000000000000000000,0.3236361655605233352311955711)
1839
(2.060000000000000000000000000,0.3325637059018157947808852386)
1840
(2.080000000000000000000000000,0.3414376680646639061197502741)
1841
(2.100000000000000000000000000,0.3502542628853236001868016155)
1842
(2.120000000000000000000000000,0.3590099958641800561590573025)
1843
(2.140000000000000000000000000,0.3677016524779047582002084146)
1844
(2.160000000000000000000000000,0.3763262838990468970337686712)
1845
(2.180000000000000000000000000,0.3848811931511300780276354433)
1846
(2.200000000000000000000000000,0.3933639217203174874682380442)
1847
(2.220000000000000000000000000,0.4017722366386150185801703062)
1848
(2.240000000000000000000000000,0.4101041180483072938071798599)
1849
(2.260000000000000000000000000,0.4183577472527792152975220295)
1850
(2.280000000000000000000000000,0.4265314952549865109567053478)
1851
(2.300000000000000000000000000,0.4346239117815307688696096270)
1852
(2.320000000000000000000000000,0.4426337147875023718864778732)
1853
(2.340000000000000000000000000,0.4505597804349194165848217499)
1854
(2.360000000000000000000000000,0.4584011335356586677964917621)
1855
(2.380000000000000000000000000,0.4661569384481976323856783780)
1856
(2.400000000000000000000000000,0.4738264904162215078882895996)
1857
(2.420000000000000000000000000,0.4814092073361560479497626593)
1858
(2.440000000000000000000000000,0.4889046219399322863990937908)
1859
(2.460000000000000000000000000,0.4963123743787402464988274434)
1860
(2.480000000000000000000000000,0.5036322051931582467284077354)
1861
(2.500000000000000000000000000,0.5108639486548272542604836536)
1862
(2.520000000000000000000000000,0.5180075264647537445144708387)
1863
(2.540000000000000000000000000,0.5250629417933500093972620327)
1864
(2.560000000000000000000000000,0.5320302736474403848513098340)
1865
(2.580000000000000000000000000,0.5389096715496600435683203816)
1866
(2.600000000000000000000000000,0.5457013505159362590105184873)
1867
(2.620000000000000000000000000,0.5524055863170584792543541452)
1868
(2.640000000000000000000000000,0.5590227110107027207717825134)
1869
(2.660000000000000000000000000,0.5655531087306685961213413424)
1870
(2.680000000000000000000000000,0.5719972117205058042214602911)
1871
(2.700000000000000000000000000,0.5783554965991442740802876900)
1872
(2.720000000000000000000000000,0.5846284808465924416813272699)
1873
(2.740000000000000000000000000,0.5908167194982262720959959025)
1874
(2.760000000000000000000000000,0.5969208020366532749488519901)
1875
(2.780000000000000000000000000,0.6029413494705972191016122489)
1876
(2.800000000000000000000000000,0.6088790115907074308408645888)
1877
(2.820000000000000000000000000,0.6147344643926488697994771583)
1878
(2.840000000000000000000000000,0.6205084076582734789474214744)
1879
(2.860000000000000000000000000,0.6262015626861078540429399870)
1880
(2.880000000000000000000000000,0.6318146701628156731049755876)
1881
(2.900000000000000000000000000,0.6373484881677044668307021565)
1882
(2.920000000000000000000000000,0.6428037903027443557826033937)
1883
(2.940000000000000000000000000,0.6481813639409507139323591218)
1884
(2.960000000000000000000000000,0.6534820085863529187302520901)
1885
(2.980000000000000000000000000,0.6587065343391271590629544240)
1886
(3.000000000000000000000000000,0.6638557604598125793076126510)
1887
1888
\psline[linecolor=green]
1889
(0.01999999999999999999999999999,-1.910549818178972717028934024)
1890
(0.03999999999999999999999999999,-3.328293754455585432531307324)
1891
(0.05999999999999999999999999999,-4.342466750788308649246155678)
1892
(0.07999999999999999999999999999,-5.028915776566633634629218476)
1893
(0.09999999999999999999999999999,-5.451829867918115376983818819)
1894
(0.1199999999999999999999999999,-5.665280766670304979484911757)
1895
(0.1399999999999999999999999999,-5.714589819284558275164336590)
1896
(0.1599999999999999999999999999,-5.637536427188076075001910574)
1897
(0.1799999999999999999999999999,-5.465422751584693225497220410)
1898
(0.1999999999999999999999999999,-5.224008635206118416756213440)
1899
(0.2199999999999999999999999999,-4.934329863946300711496384733)
1900
(0.2399999999999999999999999999,-4.613411994855483084402593901)
1901
(0.2599999999999999999999999999,-4.274891056111909740526739604)
1902
(0.2799999999999999999999999999,-3.929551504198051917653241983)
1903
(0.2999999999999999999999999999,-3.585790922181544082824074714)
1904
(0.3199999999999999999999999999,-3.250020074271328283401893598)
1905
(0.3399999999999999999999999999,-2.927006105154810291785843069)
1906
(0.3599999999999999999999999999,-2.620165894256366920568154652)
1907
(0.3799999999999999999999999999,-2.331815848679017663593063292)
1908
(0.3999999999999999999999999999,-2.063383745906609209929736921)
1909
(0.4199999999999999999999999999,-1.815587618543642244529960790)
1910
(0.4399999999999999999999999999,-1.588586107504827895129290510)
1911
(0.4599999999999999999999999999,-1.382104195353979974512854591)
1912
(0.4799999999999999999999999999,-1.195537765561407944364102925)
1913
(0.5000000000000000000000000000,-1.028040013537872770444407419)
1914
(0.5199999999999999999999999999,-0.8785923584109288988127809983)
1915
(0.5399999999999999999999999999,-0.7460621675182194230509403478)
1916
(0.5599999999999999999999999999,-0.6292493053591909031138137943)
1917
(0.5799999999999999999999999999,-0.5269232521736847868578511353)
1918
(0.5999999999999999999999999999,-0.4378523014009181016642273025)
1919
(0.6199999999999999999999999999,-0.3608261371459462024790731537)
1920
(0.6400000000000000000000000000,-0.2946729097303626103170188882)
1921
(0.6600000000000000000000000000,-0.2382717668896532168542924924)
1922
(0.6800000000000000000000000000,-0.1905616578415173212845347170)
1923
(0.6999999999999999999999999999,-0.1505471051104702859570336805)
1924
(0.7200000000000000000000000000,-0.1173015326577337443710408012)
1925
(0.7400000000000000000000000000,-0.08996864671100515617835437029)
1926
(0.7600000000000000000000000000,-0.06776228606367044028206463430)
1927
(0.7800000000000000000000000000,-0.04996509002427872773217379098)
1928
(0.8000000000000000000000000000,-0.03592627330108558665817967469)
1929
(0.8200000000000000000000000000,-0.02505874669880364598032078822)
1930
(0.8400000000000000000000000000,-0.01683577950957875130761538250)
1931
(0.8600000000000000000000000000,-0.01078736293991867023125405031)
1932
(0.8800000000000000000000000000,-0.006496402979961485124027268553)
1933
(0.9000000000000000000000000000,-0.003594845039201646547761030967)
1934
(0.9200000000000000000000000000,-0.001759810780368766619778958737)
1935
(0.9400000000000000000000000000,-0.0007098092972281847125800067572)
1936
(0.9600000000000000000000000000,-0.0002010695909836339875807547281)
1937
(0.9800000000000000000000000000,-0.00002402875616666983635585211570)
1938
(1.000000000000000000000000000,0)
1939
(1.020000000000000000000000000,0.00002196375039308742969529948158)
1940
(1.040000000000000000000000000,0.0001680016455258650783799842746)
1941
(1.060000000000000000000000000,0.0005421684876389917271685852040)
1942
(1.080000000000000000000000000,0.001228950684518991100489837915)
1943
(1.100000000000000000000000000,0.002295578458899382426642673193)
1944
(1.120000000000000000000000000,0.003794140665526099285005377488)
1945
(1.140000000000000000000000000,0.005763510386442991213032052700)
1946
(1.160000000000000000000000000,0.008231090753972734568317815952)
1947
(1.180000000000000000000000000,0.01121439129061067038742726646)
1948
(1.200000000000000000000000000,0.01472244554054399724130824150)
1949
(1.220000000000000000000000000,0.01875708097033245260992446198)
1950
(1.240000000000000000000000000,0.02331405209650245297979054622)
1951
(1.260000000000000000000000000,0.02838404760558056129141688079)
1952
(1.280000000000000000000000000,0.03395358190910612593500758156)
1953
(1.300000000000000000000000000,0.04000578115684879891056518516)
1954
(1.320000000000000000000000000,0.04652107324407005794341223824)
1955
(1.340000000000000000000000000,0.05347779081622103654842805991)
1956
(1.360000000000000000000000000,0.06085269571553849908389491774)
1957
(1.380000000000000000000000000,0.06862143274342554115528276196)
1958
(1.400000000000000000000000000,0.07675892004198595115316291488)
1959
(1.420000000000000000000000000,0.08523968283670371437561330600)
1960
(1.440000000000000000000000000,0.09403813673692726866066612962)
1961
(1.460000000000000000000000000,0.1031288262666381520103097399)
1962
(1.480000000000000000000000000,0.1124866237985848678871665261)
1963
(1.500000000000000000000000000,0.1220868935926706116686659867)
1964
(1.520000000000000000000000000,0.1319056251959508170110667056)
1965
(1.540000000000000000000000000,0.1419195400473983158625881060)
1966
(1.560000000000000000000000000,0.1521061747457908814246658072)
1967
(1.580000000000000000000000000,0.1624439440832507944297862439)
1968
(1.600000000000000000000000000,0.1729121866193372222023561673)
1969
(1.620000000000000000000000000,0.1834911952701074546569145683)
1970
(1.640000000000000000000000000,0.1941622351119767410159809078)
1971
(1.660000000000000000000000000,0.2049075503501423190683346673)
1972
(1.680000000000000000000000000,0.2157103621743391244267414930)
1973
(1.700000000000000000000000000,0.2265548590192674089289115802)
1974
(1.720000000000000000000000000,0.2374261805616739870813226735)
1975
(1.740000000000000000000000000,0.2483103966192951761126215552)
1976
(1.760000000000000000000000000,0.2591944819672342134585977619)
1977
(1.780000000000000000000000000,0.2700662879534542623387362612)
1978
(1.800000000000000000000000000,0.2809145116755873142986228792)
1979
(1.820000000000000000000000000,0.2917286633749256839900721677)
1980
(1.840000000000000000000000000,0.3024990326090861136136968306)
1981
(1.860000000000000000000000000,0.3132166536813021915467125781)
1982
(1.880000000000000000000000000,0.3238732707305705185499973839)
1983
(1.900000000000000000000000000,0.3344613028219869998111690162)
1984
(1.920000000000000000000000000,0.3449738093196727135564861591)
1985
(1.940000000000000000000000000,0.3554044557748861399401013581)
1986
(1.960000000000000000000000000,0.3657474805185004177973961571)
1987
(1.980000000000000000000000000,0.3759976621093057385236246952)
1988
(2.000000000000000000000000000,0.3861502877569540289185208930)
1989
(2.020000000000000000000000000,0.3962011228102289889273545371)
1990
(2.040000000000000000000000000,0.4061463813771859847178675520)
1991
(2.060000000000000000000000000,0.4159826981230994588784952567)
1992
(2.080000000000000000000000000,0.4257071012746624322320603834)
1993
(2.100000000000000000000000000,0.4353169868441275603047087769)
1994
(2.120000000000000000000000000,0.4448100940747250284099953318)
1995
(2.140000000000000000000000000,0.4541844820984377137883735715)
1996
(2.160000000000000000000000000,0.4634385077887892231919324532)
1997
(2.180000000000000000000000000,0.4725708047844657391814834457)
1998
(2.200000000000000000000000000,0.4815802636541348554683526508)
1999
(2.220000000000000000000000000,0.4904660131685546587076678171)
2000
(2.240000000000000000000000000,0.4992274026428169111861037990)
2001
(2.260000000000000000000000000,0.5078639853091916125210749951)
2002
(2.280000000000000000000000000,0.5163755026794064024888511710)
2003
(2.300000000000000000000000000,0.5247618698541889497106369617)
2004
(2.320000000000000000000000000,0.5330231617374235307992890964)
2005
(2.340000000000000000000000000,0.5411596001122369451656744507)
2006
(2.360000000000000000000000000,0.5491715415366574837070107171)
2007
(2.380000000000000000000000000,0.5570594660171176277599649367)
2008
(2.400000000000000000000000000,0.5648239664189391243062282569)
2009
(2.420000000000000000000000000,0.5724657385739985475690588726)
2010
(2.440000000000000000000000000,0.5799855720469798406632659693)
2011
(2.460000000000000000000000000,0.5873843415229411786222937095)
2012
(2.480000000000000000000000000,0.5946629987803257341072098062)
2013
(2.500000000000000000000000000,0.6018225652150032122247252088)
2014
(2.520000000000000000000000000,0.6088641248824191405108564520)
2015
(2.540000000000000000000000000,0.6157888180264332562096378139)
2016
(2.560000000000000000000000000,0.6225978350649314740520108950)
2017
(2.580000000000000000000000000,0.6292924110037851254552760436)
2018
(2.600000000000000000000000000,0.6358738202521960828043210020)
2019
(2.620000000000000000000000000,0.6423433718138987115139285299)
2020
(2.640000000000000000000000000,0.6487024048300827763130509282)
2021
(2.660000000000000000000000000,0.6549522844512504205689659280)
2022
(2.680000000000000000000000000,0.6610943980165213768587586068)
2023
(2.700000000000000000000000000,0.6671301515201509823860404837)
2024
(2.720000000000000000000000000,0.6730609663462236142722385731)
2025
(2.740000000000000000000000000,0.6788882762536288503425038778)
2026
(2.760000000000000000000000000,0.6846135245945186681617571307)
2027
(2.780000000000000000000000000,0.6902381617504815185771510018)
2028
(2.800000000000000000000000000,0.6957636427716537853304021984)
2029
(2.820000000000000000000000000,0.7011914252049219569221640752)
2030
(2.840000000000000000000000000,0.7065229670982510585358779478)
2031
(2.860000000000000000000000000,0.7117597251690080067453385983)
2032
(2.880000000000000000000000000,0.7169031531249342102313090749)
2033
(2.900000000000000000000000000,0.7219547001271617194348075046)
2034
(2.920000000000000000000000000,0.7269158093853633841387150823)
2035
(2.940000000000000000000000000,0.7317879168757817193385808779)
2036
(2.960000000000000000000000000,0.7365724501734954406402076494)
2037
(2.980000000000000000000000000,0.7412708273908588482867037627)
2038
(3.000000000000000000000000000,0.7458844562145893364825688878)
2039
2040
\psline[linecolor=magenta]
2041
(0.01999999999999999999999999999,-2.104245054477689947822295362)
2042
(0.03999999999999999999999999999,-3.666399647345513358746222121)
2043
(0.05999999999999999999999999999,-4.784497655504236361822016239)
2044
(0.07999999999999999999999999999,-5.541880443178120997704830499)
2045
(0.09999999999999999999999999999,-6.009094621754927671141721822)
2046
(0.1199999999999999999999999999,-6.245581871953572776496452910)
2047
(0.1399999999999999999999999999,-6.301178096252551811555028844)
2048
(0.1599999999999999999999999999,-6.217438741422336918526547376)
2049
(0.1799999999999999999999999999,-6.028806466437941337877848423)
2050
(0.1999999999999999999999999999,-5.763636503163330512506794493)
2051
(0.2199999999999999999999999999,-5.445094124035465712584502954)
2052
(0.2399999999999999999999999999,-5.091937638079895339902467621)
2053
(0.2599999999999999999999999999,-4.719199319241108737235096710)
2054
(0.2799999999999999999999999999,-4.338775656091723327350327752)
2055
(0.2999999999999999999999999999,-3.959937319551313205953198587)
2056
(0.3199999999999999999999999999,-3.589768289842191168634054794)
2057
(0.3399999999999999999999999999,-3.233542675661699365574078417)
2058
(0.3599999999999999999999999999,-2.895046904074949217664097264)
2059
(0.3799999999999999999999999999,-2.576854162785900688499447557)
2060
(0.3999999999999999999999999999,-2.280557238943954287221700716)
2061
(0.4199999999999999999999999999,-2.006965220567077145427560642)
2062
(0.4399999999999999999999999999,-1.756268906864516164261265205)
2063
(0.4599999999999999999999999999,-1.528179210182520096002654127)
2064
(0.4799999999999999999999999999,-1.322042322308378543937895330)
2065
(0.5000000000000000000000000000,-1.136934958369686047728745606)
2066
(0.5199999999999999999999999999,-0.9717425792272044572008146201)
2067
(0.5399999999999999999999999999,-0.8252231246329967190039910896)
2068
(0.5599999999999999999999999999,-0.6960584610439434601298478492)
2069
(0.5799999999999999999999999999,-0.5828954564337716587266322177)
2070
(0.5999999999999999999999999999,-0.4843783364347153875700770391)
2071
(0.6199999999999999999999999999,-0.3991737485129943548616314484)
2072
(0.6400000000000000000000000000,-0.3259897606662151757888194374)
2073
(0.6600000000000000000000000000,-0.2635898455429055945027949757)
2074
(0.6800000000000000000000000000,-0.2108027473415827270214627494)
2075
(0.6999999999999999999999999999,-0.1665289949674682092543372954)
2076
(0.7200000000000000000000000000,-0.1297447085265767260207484148)
2077
(0.7400000000000000000000000000,-0.09950324532979773141884766191)
2078
(0.7600000000000000000000000000,-0.07493514435879864287888759449)
2079
(0.7800000000000000000000000000,-0.05524675298083382350237112469)
2080
(0.8000000000000000000000000000,-0.03971785512430462894185440937)
2081
(0.8200000000000000000000000000,-0.02769856482685600460591471401)
2082
(0.8400000000000000000000000000,-0.01860570186912848753879660113)
2083
(0.8600000000000000000000000000,-0.01191882606519529829789310535)
2084
(0.8800000000000000000000000000,-0.007176072768101508285542491937)
2085
(0.9000000000000000000000000000,-0.003969903445604839492555209036)
2086
(0.9200000000000000000000000000,-0.001942861063700844162211085040)
2087
(0.9400000000000000000000000000,-0.0007833998473834845254048995470)
2088
(0.9600000000000000000000000000,-0.0002218422110628764979778425452)
2089
(0.9800000000000000000000000000,-0.00002650177671376196752112723741)
2090
(1.000000000000000000000000000,0)
2091
(1.020000000000000000000000000,0.00002420536753511631102906178318)
2092
(1.040000000000000000000000000,0.0001850692604556279737743133970)
2093
(1.060000000000000000000000000,0.0005969810089215264353797316660)
2094
(1.080000000000000000000000000,0.001352558797285380101946935102)
2095
(1.100000000000000000000000000,0.002525220573845138708143405647)
2096
(1.120000000000000000000000000,0.004171537914022092627588001132)
2097
(1.140000000000000000000000000,0.006333381411207529409140359950)
2098
(1.160000000000000000000000000,0.009039867642872384880470492163)
2099
(1.180000000000000000000000000,0.01230911874444948741641930550)
2100
(1.200000000000000000000000000,0.01614984621331312341913050302)
2101
(1.220000000000000000000000000,0.02056277083865422320265761913)
2102
(1.240000000000000000000000000,0.02554189067636591107908141471)
2103
(1.260000000000000000000000000,0.03107560881658074781144565017)
2104
(1.280000000000000000000000000,0.03714773237132557616677506502)
2105
(1.300000000000000000000000000,0.04373835367903342486782565108)
2106
(1.320000000000000000000000000,0.05082462421275766114879731782)
2107
(1.340000000000000000000000000,0.05838143111550266898804094415)
2108
(1.360000000000000000000000000,0.06638198568987699222574031014)
2109
(1.380000000000000000000000000,0.07479833255693715183435668524)
2110
(1.400000000000000000000000000,0.08360178758381593021108905768)
2111
(1.420000000000000000000000000,0.09276331207181913309718784668)
2112
(1.440000000000000000000000000,0.1022538301040232027045592523)
2113
(1.460000000000000000000000000,0.1120444953799117452545269812)
2114
(1.480000000000000000000000000,0.1221069133185093921244768397)
2115
(1.500000000000000000000000000,0.1324133236937214788659146620)
2116
(1.520000000000000000000000000,0.1429367485779942448221529004)
2117
(1.540000000000000000000000000,0.1536511099139245694654572725)
2118
(1.560000000000000000000000000,0.1645313206083280796881014150)
2119
(1.580000000000000000000000000,0.1755533526492333058963654287)
2120
(1.600000000000000000000000000,0.1866942853826025691479519951)
2121
(1.620000000000000000000000000,0.1979323367512682113245813686)
2122
(1.640000000000000000000000000,0.2092468799923677291856314780)
2123
(1.660000000000000000000000000,0.2206184480100569018237728052)
2124
(1.680000000000000000000000000,0.2320287273859680953043430904)
2125
(1.700000000000000000000000000,0.2434605437591978339335276225)
2126
(1.720000000000000000000000000,0.2548978400989704337327178666)
2127
(1.740000000000000000000000000,0.2663256492049593856423508978)
2128
(1.760000000000000000000000000,0.2777300616010131620393334353)
2129
(1.780000000000000000000000000,0.2890981898362333460842426785)
2130
(1.800000000000000000000000000,0.3004181300715570841648201730)
2131
(1.820000000000000000000000000,0.3116789217088385569912253906)
2132
(1.840000000000000000000000000,0.3228705057116158817591711646)
2133
(1.860000000000000000000000000,0.3339836821710789566181805254)
2134
(1.880000000000000000000000000,0.3450100675860878475745392264)
2135
(1.900000000000000000000000000,0.3559420522513770614965380987)
2136
(1.920000000000000000000000000,0.3667727580823428098123093523)
2137
(1.940000000000000000000000000,0.3774959971471479623065700634)
2138
(1.960000000000000000000000000,0.3881062311264651907589883785)
2139
(1.980000000000000000000000000,0.3985985318772543719872673261)
2140
(2.000000000000000000000000000,0.4089685432388427433489115742)
2141
(2.020000000000000000000000000,0.4192124441866143288636440934)
2142
(2.040000000000000000000000000,0.4293269134102452516434951821)
2143
(2.060000000000000000000000000,0.4393090953691239654586144173)
2144
(2.080000000000000000000000000,0.4491565678569003079917095419)
2145
(2.100000000000000000000000000,0.4588673110895908313957009944)
2146
(2.120000000000000000000000000,0.4684396783169487801751232604)
2147
(2.140000000000000000000000000,0.4778723679445429992902405487)
2148
(2.160000000000000000000000000,0.4871643971438742541505610947)
2149
(2.180000000000000000000000000,0.4963150769196157473989032353)
2150
(2.200000000000000000000000000,0.5053239885964524505991632600)
2151
(2.220000000000000000000000000,0.5141909616827935363213185997)
2152
(2.240000000000000000000000000,0.5229160530646503461141343847)
2153
(2.260000000000000000000000000,0.5314995274800376099733243895)
2154
(2.280000000000000000000000000,0.5399418392222165272756865963)
2155
(2.300000000000000000000000000,0.5482436150188211610498313798)
2156
(2.320000000000000000000000000,0.5564056380332767481699267672)
2157
(2.340000000000000000000000000,0.5644288329348267126642593505)
2158
(2.360000000000000000000000000,0.5723142519838439598978481143)
2159
(2.380000000000000000000000000,0.5800630620798324658967221504)
2160
(2.400000000000000000000000000,0.5876765327205585154120857722)
2161
(2.420000000000000000000000000,0.5951560248220275259300914382)
2162
(2.440000000000000000000000000,0.6025029803504906259777054363)
2163
(2.460000000000000000000000000,0.6097189127192805778749317858)
2164
(2.480000000000000000000000000,0.6168053979050011010627388718)
2165
(2.500000000000000000000000000,0.6237640662393945814028236482)
2166
(2.520000000000000000000000000,0.6305965948350628614998520806)
2167
(2.540000000000000000000000000,0.6373047006050909149467564465)
2168
(2.560000000000000000000000000,0.6438901338385040985335445982)
2169
(2.580000000000000000000000000,0.6503546722953600277340798773)
2170
(2.600000000000000000000000000,0.6567001157871224776080108597)
2171
(2.620000000000000000000000000,0.6629282812097761116990888823)
2172
(2.640000000000000000000000000,0.6690409979989084829860440872)
2173
(2.660000000000000000000000000,0.6750401039777026951819632792)
2174
(2.680000000000000000000000000,0.6809274415704450251247746277)
2175
(2.700000000000000000000000000,0.6867048543557527270190807500)
2176
(2.720000000000000000000000000,0.6923741839352653779659473982)
2177
(2.740000000000000000000000000,0.6979372670950166868106093773)
2178
(2.760000000000000000000000000,0.7033959332381117187044241598)
2179
(2.780000000000000000000000000,0.7087520020686767323695658151)
2180
(2.800000000000000000000000000,0.7140072815083256163198846647)
2181
(2.820000000000000000000000000,0.7191635658275990555087225311)
2182
(2.840000000000000000000000000,0.7242226339759812636761073316)
2183
(2.860000000000000000000000000,0.7291862480951858956582109533)
2184
(2.880000000000000000000000000,0.7340561522014293719840637414)
2185
(2.900000000000000000000000000,0.7388340710233782591027649232)
2186
(2.920000000000000000000000000,0.7435217089833696474540397469)
2187
(2.940000000000000000000000000,0.7481207493103618505066630155)
2188
(2.960000000000000000000000000,0.7526328532738794689898061792)
2189
(2.980000000000000000000000000,0.7570596595289742181710614400)
2190
(3.000000000000000000000000000,0.7614027835629332038249116280)
2191
2192
\rput(.2,-2.35){$E_1$}
2193
\rput(-.05,-5.5){$E_2$}
2194
\rput(.33,-6){$E_3$}
2195
\rput(1.25,1.8){$E_1=[0,0,1,-7,6],\ E_2=[1,-1,1,-6,0],\ E_3=[1,-1,0,-16,28]$}
2196
\pscircle[linecolor=red](1,0){0.1}
2197
\end{pspicture}
2198
\caption{Curves of rank 3}\label{curvesof3}
2199
\begin{pspicture}(0,0)(2,2)
2200
\end{pspicture}
2201
\end{center}
2202
\end{figure}
2203
%\vspace{10cm}
2204
2205
Note that as in the case of the curves of rank 2, the graphs are arranged according to conductor. In this case, a curve of higher conductor is always less than a curve of lower conductor. One might be led to turn this observation into a conjecture, and in fact we were at first going to do so. However, closer examination of the evidence shows that while it may sometimes be true, it is not always the case that if the conductor of $E_1$ is larger than that of $E_2$ then $|L(E_1,x)| > |L(E_2,x)|$, as might be at first believed. One counter example of this can be seen with two curves of rank 2: Take $E_1$ to be $[1,1,1,-15,16]$ and $E_2$ to be $[0,1,1,-4,2]$. Then both have rank 2 and the conductor of $E_1$ is 563 and the conductor of $E_2$ is 571. However $L(E_1,0.5)=0.34614\ldots$ while $L(E_2,0.5)=0.27975\ldots$. Hence the possible conjecture is untrue.
2206
2207
One can also see that the possible conjecture would not hold for curves of rank 4. This can be seen in Figure~\ref{curvesof4} which is a graph of the curves $[0,1,1,-72,210],\ [0,0,1,-7,36],\ [1,0,0,-202,1089],\ {\rm and }\ [0,1,1,-2,42]$.
2208
\begin{figure}
2209
\begin{center}
2210
\psset{unit=1.3in}
2211
\begin{pspicture}(0,-.5)(1.5,1.5)
2212
\psgrid[gridcolor=gray]
2213
2214
% axes
2215
\psline[linewidth=0.03]{->}(0,0)(1.5,0)\rput(1.6,0){$x$}
2216
\psline[linewidth=0.03]{->}(0,-.5)(0,1.5)\rput(0,1.6){$y$}
2217
2218
\psline[linecolor=blue]
2219
(0.5899999999999999999999999999,1.558791254529780353650676198)
2220
(0.5999999999999999999999999999,1.351988705682560324024993404)
2221
(0.6099999999999999999999999999,1.169592260303107032062926208)
2222
(0.6199999999999999999999999999,1.009068246962403403551644221)
2223
(0.6299999999999999999999999999,0.8681101284496229585018279966)
2224
(0.6399999999999999999999999999,0.7446217734728785869074503599)
2225
(0.6499999999999999999999999999,0.6367015486567037663083672278)
2226
(0.6599999999999999999999999999,0.5426272496184963526095489266)
2227
(0.6699999999999999999999999999,0.4608418773415704591772273263)
2228
(0.6799999999999999999999999999,0.3899402557982345166679019158)
2229
(0.6899999999999999999999999999,0.3286564784879022775320534447)
2230
(0.6999999999999999999999999999,0.2758521649583901602674067639)
2231
(0.7099999999999999999999999999,0.2305055032246879731540253230)
2232
(0.7199999999999999999999999999,0.1917010500716215995191664790)
2233
(0.7299999999999999999999999999,0.1586202583358842489815837760)
2234
(0.7399999999999999999999999999,0.1305326982444761799542332608)
2235
(0.7500000000000000000000000000,0.1067879385979792013102618269)
2236
(0.7599999999999999999999999999,0.08680805290476720679628680319)
2237
(0.7699999999999999999999999999,0.07008071538952592319263772414)
2238
(0.7799999999999999999999999999,0.05615285202442776536349921789)
2239
(0.7899999999999999999999999999,0.04462481228506282018879265400)
2240
(0.7999999999999999999999999999,0.03514502814821230607440259138)
2241
(0.8099999999999999999999999999,0.02740512786715263153885193895)
2242
(0.8199999999999999999999999999,0.02113547323345781856292592246)
2243
(0.8299999999999999999999999999,0.01610109032084547909214109005)
2244
(0.8399999999999999999999999999,0.01209796507167926801966628220)
2245
(0.8499999999999999999999999999,0.008949676501194541671101172961)
2246
(0.8599999999999999999999999999,0.006504341734187109035956275786)
2247
(0.8699999999999999999999999999,0.004631848533894776202955552771)
2248
(0.8799999999999999999999999999,0.003221352416877584272788557245)
2249
(0.8899999999999999999999999999,0.002179016857788496056887948057)
2250
(0.8999999999999999999999999999,0.001425976463644170233320280997)
2251
(0.9099999999999999999999999999,0.0008965043304795137617021380262)
2252
(0.9199999999999999999999999999,0.0005363660799790137332026844096)
2253
(0.9299999999999999999999999999,0.0003013443053542895298546371338)
2254
(0.9399999999999999999999999999,0.0001559183313027799352075198925)
2255
(0.9499999999999999999999999999,0.00007208531042093785217898901185)
2256
(0.9599999999999999999999999999,0.00002830973700536766548523716064)
2257
(0.9699999999999999999999999999,0.000008589458599847140722839926272)
2258
(0.9799999999999999999999999999,0.000001627206425387321048558339091)
2259
(0.9899999999999999999999999999,0.00000009754897602399621950131267372)
2260
(1.000000000000000000000000000,0)
2261
(1.009999999999999999999999999,0.00000008978456095223987548057059483)
2262
(1.019999999999999999999999999,0.000001378486876547995567563314712)
2263
(1.029999999999999999999999999,0.000006697473329868720807123775000)
2264
(1.039999999999999999999999999,0.00002031760573056346923252168770)
2265
(1.049999999999999999999999999,0.00004761933593248946337188859654)
2266
(1.059999999999999999999999999,0.00009480780569302452837525945372)
2267
(1.069999999999999999999999999,0.0001686680675922335928144394138)
2268
(1.079999999999999999999999999,0.0002763559963024442761155622950)
2269
(1.089999999999999999999999999,0.0004252208768536349134091845660)
2270
(1.099999999999999999999999999,0.0006226560400622001045201369660)
2271
(1.109999999999999999999999999,0.0008759742671938714268793882805)
2272
(1.119999999999999999999999999,0.001192305008348758134520346257)
2273
(1.129999999999999999999999999,0.001578510754032673816190363782)
2274
(1.140000000000000000000000000,0.002041120168866436488453833427)
2275
(1.149999999999999999999999999,0.002586275842240254446886288227)
2276
(1.160000000000000000000000000,0.003219694734703050150967593235)
2277
(1.170000000000000000000000000,0.003946639602648692308828792951)
2278
(1.180000000000000000000000000,0.004771899868988173826856331295)
2279
(1.190000000000000000000000000,0.005699780575449562578773716106)
2280
(1.199999999999999999999999999,0.006734098204304388767500625973)
2281
(1.210000000000000000000000000,0.007878182294968763536597674286)
2282
(1.220000000000000000000000000,0.009134881905272372183618321618)
2283
(1.230000000000000000000000000,0.01050657607934819265051981201)
2284
(1.240000000000000000000000000,0.01199518758511094815099178839)
2285
(1.250000000000000000000000000,0.01360219927512830995561689624)
2286
(1.260000000000000000000000000,0.01532867250623983928357066649)
2287
(1.270000000000000000000000000,0.01717526712637135166045155408)
2288
(1.280000000000000000000000000,0.01914226260239008384508620775)
2289
(1.290000000000000000000000000,0.02122957992125236721349782284)
2290
(1.300000000000000000000000000,0.02343680394875815559921320287)
2291
(1.310000000000000000000000000,0.02576320597654109193590344729)
2292
(1.320000000000000000000000000,0.02820776622903535691686055196)
2293
(1.330000000000000000000000000,0.03076919613857235097199719529)
2294
(1.340000000000000000000000000,0.03344596022893002974380955730)
2295
(1.350000000000000000000000000,0.03623629747600487341445468864)
2296
(1.360000000000000000000000000,0.03913824203918404832119345385)
2297
(1.370000000000000000000000000,0.04214964327881265808006302623)
2298
(1.380000000000000000000000000,0.04526818499419630401942095651)
2299
(1.390000000000000000000000000,0.04849140383314199546964241991)
2300
(1.400000000000000000000000000,0.05181670683838383643560797942)
2301
(1.410000000000000000000000000,0.05524138810860260267809311584)
2302
(1.420000000000000000000000000,0.05876264456234669447957166376)
2303
(1.430000000000000000000000000,0.06237759080219187616206276309)
2304
(1.440000000000000000000000000,0.06608327308411576756717138635)
2305
(1.450000000000000000000000000,0.06987668240347010447483765664)
2306
(1.460000000000000000000000000,0.07375476671425347577482914938)
2307
(1.470000000000000000000000000,0.07771444230274935597739127151)
2308
(1.480000000000000000000000000,0.08175260434011546664194079179)
2309
(1.490000000000000000000000000,0.08586613664129557411491288962)
2310
(1.500000000000000000000000000,0.09005192065976766427401709128)
2311
2312
\psline[linecolor=green]
2313
(0.5899999999999999999999999999,1.408951738645349791068825739)
2314
(0.5999999999999999999999999999,1.220576497569384035329874899)
2315
(0.6099999999999999999999999999,1.054670056527229920658604960)
2316
(0.6199999999999999999999999999,0.9088650127288872701058127193)
2317
(0.6299999999999999999999999999,0.7810107706029968960518767635)
2318
(0.6399999999999999999999999999,0.6691570218008401799252173492)
2319
(0.6499999999999999999999999999,0.5715381037282196868965816150)
2320
(0.6599999999999999999999999999,0.4865582440060375151751736414)
2321
(0.6699999999999999999999999999,0.4127776870214232035323863743)
2322
(0.6799999999999999999999999999,0.3488996896841461813926427941)
2323
(0.6899999999999999999999999999,0.2937583663274744903739893158)
2324
(0.6999999999999999999999999999,0.2463073571091864468768353155)
2325
(0.7099999999999999999999999999,0.2056092900337761429760971102)
2326
(0.7199999999999999999999999999,0.1708260036195251391586118659)
2327
(0.7299999999999999999999999999,0.1412094950905957248475430923)
2328
(0.7399999999999999999999999999,0.1160935576258665629263593542)
2329
(0.7500000000000000000000000000,0.09488606950579357747371977961)
2330
(0.7599999999999999999999999999,0.07706189784807905313757547181)
2331
(0.7699999999999999999999999999,0.06215637991098423888273708647)
2332
(0.7799999999999999999999999999,0.04975934558292687375528321149)
2333
(0.7899999999999999999999999999,0.03950964559450919847453214007)
2334
(0.7999999999999999999999999999,0.03109015112138070621143505461)
2335
(0.8099999999999999999999999999,0.02422319174008273935592521955)
2336
(0.8199999999999999999999999999,0.01866640010938275600277465808)
2337
(0.8299999999999999999999999999,0.01420893323900482726308204129)
2338
(0.8399999999999999999999999999,0.01066804174482323991073484858)
2339
(0.8499999999999999999999999999,0.007885960048671352850430742219)
2340
(0.8599999999999999999999999999,0.005727092040773737403995618423)
2341
(0.8699999999999999999999999999,0.004075468266300410569629984996)
2342
(0.8799999999999999999999999999,0.002832452210970140332193381404)
2343
(0.8899999999999999999999999999,0.001914674733229187665329496330)
2344
(0.8999999999999999999999999999,0.001252177114025735896145080692)
2345
(0.9099999999999999999999999999,0.0007867445634168821277088365255)
2346
(0.9199999999999999999999999999,0.0004704133317836786088012344559)
2347
(0.9299999999999999999999999999,0.0002641358193711637480827221169)
2348
(0.9399999999999999999999999999,0.0001365892595267978759182320396)
2349
(0.9499999999999999999999999999,0.00006311466770986015975577396230)
2350
(0.9599999999999999999999999999,0.00002477380024354722788835301915)
2351
(0.9699999999999999999999999999,0.000007512854708538505972098722940)
2352
(0.9799999999999999999999999999,0.000001422569193653599050211760593)
2353
(0.9899999999999999999999999999,0.00000008524210605351724452442930174)
2354
(1.000000000000000000000000000,0)
2355
(1.009999999999999999999999999,0.00000007839024568263872176265514643)
2356
(1.019999999999999999999999999,0.000001203070830731813804159530787)
2357
(1.029999999999999999999999999,0.000005843013777470149506751275867)
2358
(1.039999999999999999999999999,0.00001771923423895016891779474031)
2359
(1.049999999999999999999999999,0.00004151560699618959375006135412)
2360
(1.059999999999999999999999999,0.00008262983849455276943097989712)
2361
(1.069999999999999999999999999,0.0001469601283664755039799399535)
2362
(1.079999999999999999999999999,0.0002407234821680998197424054769)
2363
(1.089999999999999999999999999,0.0003703020293098861190414265961)
2364
(1.099999999999999999999999999,0.0005421140592981479489704421441)
2365
(1.109999999999999999999999999,0.0007625068177401088245567952622)
2366
(1.119999999999999999999999999,0.001037668403309751522152567120)
2367
(1.129999999999999999999999999,0.001373556380127490445557933172)
2368
(1.140000000000000000000000000,0.001775840968765357069425082471)
2369
(1.149999999999999999999999999,0.002249860905231972059949411758)
2370
(1.160000000000000000000000000,0.002800590262589083442124289233)
2371
(1.170000000000000000000000000,0.003432614715966637025523955206)
2372
(1.180000000000000000000000000,0.004150115900233336948097720455)
2373
(1.190000000000000000000000000,0.004956862661899177046670952895)
2374
(1.199999999999999999999999999,0.005856208144331524318022943739)
2375
(1.210000000000000000000000000,0.006851091769318447477306710927)
2376
(1.220000000000000000000000000,0.007944045289583234067420990212)
2377
(1.230000000000000000000000000,0.009137202187127706406086662957)
2378
(1.240000000000000000000000000,0.01043230978226298747910998277)
2379
(1.250000000000000000000000000,0.01183074349880196163715728593)
2380
(1.260000000000000000000000000,0.01333352280299272461674398102)
2381
(1.270000000000000000000000000,0.01494132839815387359506114084)
2382
(1.280000000000000000000000000,0.01665452031435406141927033229)
2383
(1.290000000000000000000000000,0.01847315658352397729347102944)
2384
(1.300000000000000000000000000,0.02039701223570762546399267061)
2385
(1.310000000000000000000000000,0.02242559839230877338671575739)
2386
(1.320000000000000000000000000,0.02455818126767724135196306930)
2387
(1.330000000000000000000000000,0.02679380092167350119363561138)
2388
(1.340000000000000000000000000,0.02913128963337303107880873732)
2389
(1.350000000000000000000000000,0.03156928979021034277651677293)
2390
(1.360000000000000000000000000,0.03410627120796792156440644872)
2391
(1.370000000000000000000000000,0.03674054781540667951670395372)
2392
(1.380000000000000000000000000,0.03947029365330150563217842977)
2393
(1.390000000000000000000000000,0.04229355815145050415910741518)
2394
(1.400000000000000000000000000,0.04520828065910701785320029211)
2395
(1.410000000000000000000000000,0.04821230421445417229109202265)
2396
(1.420000000000000000000000000,0.05130338854739619723193663381)
2397
(1.430000000000000000000000000,0.05447922231725384170599257843)
2398
(1.440000000000000000000000000,0.05773743459308004484849401012)
2399
(1.450000000000000000000000000,0.06107560558939802475257012154)
2400
(1.460000000000000000000000000,0.06449127667433402313293610413)
2401
(1.470000000000000000000000000,0.06798195967048487650355713075)
2402
(1.480000000000000000000000000,0.07154514547152822565322334426)
2403
(1.490000000000000000000000000,0.07517831199964155277373984655)
2404
(1.500000000000000000000000000,0.07887893153032657308207594288)
2405
2406
\psline[linecolor=cyan]
2407
(0.5899999999999999999999999999,2.199015588665317709413711570)
2408
(0.5999999999999999999999999999,1.906016586620804426533910370)
2409
(0.6099999999999999999999999999,1.647781831212067114531513502)
2410
(0.6199999999999999999999999999,1.420677981026817699365616236)
2411
(0.6299999999999999999999999999,1.221400283224710635495161491)
2412
(0.6399999999999999999999999999,1.046948052718177120911887950)
2413
(0.6499999999999999999999999999,0.8946013928165076870183563966)
2414
(0.6599999999999999999999999999,0.7618991783309902825275983718)
2415
(0.6699999999999999999999999999,0.6466183044468123178940226646)
2416
(0.6799999999999999999999999999,0.5467541902912295566120508955)
2417
(0.6899999999999999999999999999,0.4605025145927931215442473542)
2418
(0.6999999999999999999999999999,0.3862421517184161750805019909)
2419
(0.7099999999999999999999999999,0.3225192693316840112867850110)
2420
(0.7199999999999999999999999999,0.2680325436215770294350622185)
2421
(0.7299999999999999999999999999,0.2216194442313071499460678996)
2422
(0.7399999999999999999999999999,0.1822435384341253063867772779)
2423
(0.7500000000000000000000000000,0.1489827625505070104786317012)
2424
(0.7599999999999999999999999999,0.1210186079008857556815427094)
2425
(0.7699999999999999999999999999,0.09762616858653697885696754116)
2426
(0.7799999999999999999999999999,0.07816499895637797482192613084)
2427
(0.7899999999999999999999999999,0.06207072963631617339104777368)
2428
(0.7999999999999999999999999999,0.04884739237385991061111319122)
2429
(0.8099999999999999999999999999,0.03806040560194420473908878894)
2430
(0.8199999999999999999999999999,0.02933017448283379793014029190)
2431
(0.8299999999999999999999999999,0.02232626119699409529369735822)
2432
(0.8399999999999999999999999999,0.01676208334390174467704782677)
2433
(0.8499999999999999999999999999,0.01239010048104882505405604987)
2434
(0.8599999999999999999999999999,0.008997451010121706569830389596)
2435
(0.8699999999999999999999999999,0.006402003797864384336395871598)
2436
(0.8799999999999999999999999999,0.004448791071074945111642964995)
2437
(0.8899999999999999999999999999,0.003006791232638921249209160275)
2438
(0.8999999999999999999999999999,0.001966032294419271470594814845)
2439
(0.9099999999999999999999999999,0.001234988602407620086368684650)
2440
(0.9199999999999999999999999999,0.0007382454317655510514527628652)
2441
(0.9299999999999999999999999999,0.0004144078485453100709561064841)
2442
(0.9399999999999999999999999999,0.0002142319672221482626065751913)
2443
(0.9499999999999999999999999999,0.00009895837656329271402569804708)
2444
(0.9599999999999999999999999999,0.00003882906000773207935336005205)
2445
(0.9699999999999999999999999999,0.00001177060093884787890667202656)
2446
(0.9799999999999999999999999999,0.000002227839187113855536669222434)
2447
(0.9899999999999999999999999999,0.0000001334346988228414814408457764)
2448
(1.000000000000000000000000000,0)
2449
(1.009999999999999999999999999,0.0000001225877481143058962182712792)
2450
(1.019999999999999999999999999,0.000001880366488689112990040772779)
2451
(1.029999999999999999999999999,0.000009127290132682428517891920786)
2452
(1.039999999999999999999999999,0.00002766246820179264996888895463)
2453
(1.049999999999999999999999999,0.00006477177819143089727592956979)
2454
(1.059999999999999999999999999,0.0001288330321411374656731913938)
2455
(1.069999999999999999999999999,0.0002289777203297337925105182635)
2456
(1.079999999999999999999999999,0.0003748030095262305744987975910)
2457
(1.089999999999999999999999999,0.0005761282749468707169633259868)
2458
(1.099999999999999999999999999,0.0008427909974118073832579352559)
2459
(1.109999999999999999999999999,0.001184477363461872907510494601)
2460
(1.119999999999999999999999999,0.001610583369556277269409295151)
2461
(1.129999999999999999999999999,0.002130102654956271324327362566)
2462
(1.140000000000000000000000000,0.002751537674393350308682927034)
2463
(1.149999999999999999999999999,0.003482831173861046316188696357)
2464
(1.160000000000000000000000000,0.004331315253445441564533535853)
2465
(1.170000000000000000000000000,0.005303675592461750526403348329)
2466
(1.180000000000000000000000000,0.006405928676587192741750082736)
2467
(1.190000000000000000000000000,0.007643410106325703214962137100)
2468
(1.199999999999999999999999999,0.009020772283021006463705071128)
2469
(1.210000000000000000000000000,0.01054198996463062978119122313)
2470
(1.220000000000000000000000000,0.01221037236033547667230293017)
2471
(1.230000000000000000000000000,0.01402858059241569797505252693)
2472
(1.240000000000000000000000000,0.01599864949718487304130239555)
2473
(1.250000000000000000000000000,0.01812201286554102768061867772)
2474
(1.260000000000000000000000000,0.02039953133915986229598925994)
2475
(1.270000000000000000000000000,0.02283152228171883212966392809)
2476
(1.280000000000000000000000000,0.02541779103690339358635364163)
2477
(1.290000000000000000000000000,0.02815766306732444344277987453)
2478
(1.300000000000000000000000000,0.03105001654180261946700623021)
2479
(1.310000000000000000000000000,0.03409331500360824184661832624)
2480
(1.320000000000000000000000000,0.03728563980997163516842811434)
2481
(1.330000000000000000000000000,0.04062472208421756141789802454)
2482
(1.340000000000000000000000000,0.04410797396688822168113900210)
2483
(1.350000000000000000000000000,0.04773251899180346310146728071)
2484
(1.360000000000000000000000000,0.05149522144771342416672899096)
2485
(1.370000000000000000000000000,0.05539271461652808348217073121)
2486
(1.380000000000000000000000000,0.05942142780551527855188055831)
2487
(1.390000000000000000000000000,0.06357761211375751348465095040)
2488
(1.400000000000000000000000000,0.06785736489292391481332053114)
2489
(1.410000000000000000000000000,0.07225665287938760461130855295)
2490
(1.420000000000000000000000000,0.07677133398920894780240184376)
2491
(1.430000000000000000000000000,0.08139717777979051172378017298)
2492
(1.440000000000000000000000000,0.08612988459234206554149165384)
2493
(1.450000000000000000000000000,0.09096510339790077011263998453)
2494
(1.460000000000000000000000000,0.09589844837673754075812099340)
2495
(1.470000000000000000000000000,0.1009255142667295109512930409)
2496
(1.480000000000000000000000000,0.1060418905208559457191945223)
2497
(1.490000000000000000000000000,0.1112431743175291530026109176)
2498
(1.500000000000000000000000000,0.1165249824701357162892407279)
2499
2500
\psline[linecolor=magenta]
2501
(0.5899999999999999999999999999,1.976736120722995352067939403)
2502
(0.5999999999999999999999999999,1.708517310137656064698222116)
2503
(0.6099999999999999999999999999,1.472912037051071676817264645)
2504
(0.6199999999999999999999999999,1.266395708223089258443525176)
2505
(0.6299999999999999999999999999,1.085777827718086413178509712)
2506
(0.6399999999999999999999999999,0.9281751900965545112194143688)
2507
(0.6499999999999999999999999999,0.7909866664579006801457698037)
2508
(0.6599999999999999999999999999,0.6718695671442393226815967871)
2509
(0.6699999999999999999999999999,0.5687175497897532863275184577)
2510
(0.6799999999999999999999999999,0.4796400295984897966735076612)
2511
(0.6899999999999999999999999999,0.4029430397392533935512843896)
2512
(0.6999999999999999999999999999,0.3371114831126665583824494969)
2513
(0.7099999999999999999999999999,0.2807927120862045960379709721)
2514
(0.7199999999999999999999999999,0.2327813697749277376554240594)
2515
(0.7299999999999999999999999999,0.1920054247821021345838678237)
2516
(0.7399999999999999999999999999,0.1575133307589625176201553082)
2517
(0.7500000000000000000000000000,0.1284622424860648669735877437)
2518
(0.7599999999999999999999999999,0.1041072212404938132985519575)
2519
(0.7699999999999999999999999999,0.08379136384104075640766462310)
2520
(0.7799999999999999999999999999,0.06693679182816093678745518082)
2521
(0.7899999999999999999999999999,0.05303643962814000452165748762)
2522
(0.7999999999999999999999999999,0.04164658318009448812089124813)
2523
(0.8099999999999999999999999999,0.03238005329399459273202171569)
2524
(0.8199999999999999999999999999,0.02490008089468924613600133942)
2525
(0.8299999999999999999999999999,0.01891472423901226202954567234)
2526
(0.8399999999999999999999999999,0.01417183112815356746446069556)
2527
(0.8499999999999999999999999999,0.01045449204150863428831557087)
2528
(0.8599999999999999999999999999,0.007576942964091035153954150752)
2529
(0.8699999999999999999999999999,0.005380879446166508583298116853)
2530
(0.8799999999999999999999999999,0.003732146104921803398531230744)
2531
(0.8899999999999999999999999999,0.002517768341821214207997085467)
2532
(0.8999999999999999999999999999,0.001643295497465905756684355108)
2533
(0.9099999999999999999999999999,0.001030426992833565119023892370)
2534
(0.9199999999999999999999999999,0.0006148952087445474093068744658)
2535
(0.9299999999999999999999999999,0.0003445809332703709277046945258)
2536
(0.9399999999999999999999999999,0.0001778391601789636912881492946)
2537
(0.9499999999999999999999999999,0.00008201485229827105942580041094)
2538
(0.9599999999999999999999999999,0.00003212999479461114974751801040)
2539
(0.9699999999999999999999999999,0.000009724858511306607899244887840)
2540
(0.9799999999999999999999999999,0.000001837876996266111304006945913)
2541
(0.9899999999999999999999999999,0.0000001099173930807899237151231104)
2542
(1.000000000000000000000000000,0)
2543
(1.009999999999999999999999999,0.0000001006992150570562644212050464)
2544
(1.019999999999999999999999999,0.000001542545069533539236402217209)
2545
(1.029999999999999999999999999,0.000007477744901345380860210610928)
2546
(1.039999999999999999999999999,0.00002263446417837976277825693542)
2547
(1.049999999999999999999999999,0.00005293374919206818752699680109)
2548
(1.059999999999999999999999999,0.0001051619473072022870556763628)
2549
(1.069999999999999999999999999,0.0001866921875104410743313586042)
2550
(1.079999999999999999999999999,0.0003052491297922603789371943498)
2551
(1.089999999999999999999999999,0.0004687117808649054893386103314)
2552
(1.099999999999999999999999999,0.0006849497100862949593519614355)
2553
(1.109999999999999999999999999,0.0009616884872295566536971367451)
2554
(1.119999999999999999999999999,0.001306400606689293684104795025)
2555
(1.129999999999999999999999999,0.001726218564405462836607359702)
2556
(1.140000000000000000000000000,0.002227867117551052991057291546)
2557
(1.149999999999999999999999999,0.002817612085993163644858383415)
2558
(1.160000000000000000000000000,0.003501223351613033559048005858)
2559
(1.170000000000000000000000000,0.004283949979472820041044185467)
2560
(1.180000000000000000000000000,0.005170505626066861536487136012)
2561
(1.190000000000000000000000000,0.006165062616830599318148632816)
2562
(1.199999999999999999999999999,0.007271253269864738289349521945)
2563
(1.210000000000000000000000000,0.008492177217464009721824246551)
2564
(1.220000000000000000000000000,0.009830413633361827465272540039)
2565
(1.230000000000000000000000000,0.01128803741331062836150225040)
2566
(1.240000000000000000000000000,0.01286663848127198219561953963)
2567
(1.250000000000000000000000000,0.01456734350452158084903682243)
2568
(1.260000000000000000000000000,0.01639083939969323529579580115)
2569
(1.270000000000000000000000000,0.01833739809939291135706457836)
2570
(1.280000000000000000000000000,0.02040690212660506947910781844)
2571
(1.290000000000000000000000000,0.02259887059268970952312708986)
2572
(1.300000000000000000000000000,0.02491248529524142150398640347)
2573
(1.310000000000000000000000000,0.02734661664528134463682242892)
2574
(1.320000000000000000000000000,0.02989984919993364064464282259)
2575
(1.330000000000000000000000000,0.03257050661758475914763380288)
2576
(1.340000000000000000000000000,0.03535667588815736750486532056)
2577
(1.350000000000000000000000000,0.03825623072211363197943141612)
2578
(1.360000000000000000000000000,0.04126685400864311057708480763)
2579
(1.370000000000000000000000000,0.04438605927664818858715924400)
2580
(1.380000000000000000000000000,0.04761121111202912433000484882)
2581
(1.390000000000000000000000000,0.05093954450176469468213296128)
2582
(1.400000000000000000000000000,0.05436818308971903531443321366)
2583
(1.410000000000000000000000000,0.05789415634128238382769952204)
2584
(1.420000000000000000000000000,0.06151441562414389408101760106)
2585
(1.430000000000000000000000000,0.06522584922094119242496454926)
2586
(1.440000000000000000000000000,0.06902529629645105547211471834)
2587
(1.450000000000000000000000000,0.07290955984757253509860291772)
2588
(1.460000000000000000000000000,0.07687541866878113433726616982)
2589
(1.470000000000000000000000000,0.08091963836915441830170723010)
2590
(1.480000000000000000000000000,0.08503898147962281282506996406)
2591
(1.490000000000000000000000000,0.08923021669090597980707720792)
2592
(1.500000000000000000000000000,0.09349012726376286750044670569)
2593
2594
\pscircle[linecolor=red](1,0){0.1}
2595
\end{pspicture}
2596
\caption{Curves of rank 4}\label{curvesof4}
2597
\end{center}
2598
\end{figure}
2599
All of the lines in the graph are so close it is hard to make out exactly what is going on in the graph. However, looking directly at the data we can see that the conjecture does not hold. The curve $[0,1,1,-72,210]$ has conductor 501029. At the point $x=.59$ the $L$-series has the value 1.558791254529780353650676198. The curve $[0,0,1,-7,36]$ has conductor 545723 which is larger than the first curve. However at $x=.59$ the $L$-series for this curve has the value\\ 1.408951738645349791068825739. Hence the conjecture does not hold for $N=4$.
2600
2601
With the help of some powerful computing power, we can even calculate data for curves of rank 5 or higher. An example of a graph of the $L$-series for a curve of rank 5 is shown in Figure~\ref{rank5}. The curve used for this graph is $[0,1,1,-30,390]$ which has rank 5 and conductor 67445803. It was interesting to note that the $L$-series of all of the curves of rank 5 that we graphed looked amazingly similar to the naked eye.
2602
2603
\begin{figure}
2604
\begin{center}
2605
\psset{unit=2in}
2606
\begin{pspicture}(0.5,-.1)(2,.5)
2607
\psgrid[gridcolor=gray]
2608
2609
% axes
2610
\psline[linewidth=0.015]{->}(0,0)(2,0)\rput(2.1,0){$x$}
2611
\psline[linewidth=0.015]{->}(0.4,-.3)(0.4,.7)\rput(0.4,.75){$y$}
2612
2613
\psline[linewidth=0.025, linecolor=blue]
2614
(0.5000000000000000000000000000,0)
2615
(0.5099999999999999999999999999,0)
2616
(0.5199999999999999999999999999,0)
2617
(0.5299999999999999999999999999,0)
2618
(0.5399999999999999999999999999,0)
2619
(0.5499999999999999999999999999,0)
2620
(0.5599999999999999999999999999,0)
2621
(0.5699999999999999999999999999,0)
2622
(0.5799999999999999999999999999,0)
2623
(0.5899999999999999999999999999,0)
2624
(0.5999999999999999999999999999,0)
2625
(0.6099999999999999999999999999,0)
2626
(0.6199999999999999999999999999,0)
2627
(0.6299999999999999999999999999,0)
2628
(0.6399999999999999999999999999,0)
2629
(0.6499999999999999999999999999,0)
2630
(0.6599999999999999999999999999,0)
2631
(0.6699999999999999999999999999,0)
2632
(0.6799999999999999999999999999,0)
2633
(0.6899999999999999999999999999,0)
2634
(0.6999999999999999999999999999,0)
2635
(0.7099999999999999999999999999,0)
2636
(0.7199999999999999999999999999,0)
2637
(0.7299999999999999999999999999,0)
2638
(0.7399999999999999999999999999,0)
2639
(0.7500000000000000000000000000,0)
2640
(0.7599999999999999999999999999,0)
2641
(0.7699999999999999999999999999,0)
2642
(0.7799999999999999999999999999,0)
2643
(0.7899999999999999999999999999,0)
2644
(0.7999999999999999999999999999,0)
2645
(0.8099999999999999999999999999,0)
2646
(0.8199999999999999999999999999,0)
2647
(0.8299999999999999999999999999,0)
2648
(0.8399999999999999999999999999,0)
2649
(0.8499999999999999999999999999,0)
2650
(0.8599999999999999999999999999,0)
2651
(0.8699999999999999999999999999,0)
2652
(0.8799999999999999999999999999,0)
2653
(0.8899999999999999999999999999,0)
2654
(0.8999999999999999999999999999,0)
2655
(0.9099999999999999999999999999,0)
2656
(0.9199999999999999999999999999,0)
2657
(0.9299999999999999999999999999,0)
2658
(0.9399999999999999999999999999,0)
2659
(0.9499999999999999999999999999,0)
2660
(0.9599999999999999999999999999,0)
2661
(0.9699999999999999999999999999,0)
2662
(0.9799999999999999999999999999,0)
2663
(0.9899999999999999999999999999,0)
2664
(1.000000000000000000000000000,0)
2665
(1.009999999999999999999999999,0.000000004075890711357673222624000597)
2666
(1.019999999999999999999999999,0.0000001221644719816407636735382465)
2667
(1.029999999999999999999999999,0.0000008692149279992273758358010065)
2668
(1.039999999999999999999999999,0.000003433216884218353044506884841)
2669
(1.049999999999999999999999999,0.000009823967499347080166738741575)
2670
(1.059999999999999999999999999,0.00002292898011990176495692576443)
2671
(1.069999999999999999999999999,0.00004650149735776723209834923410)
2672
(1.079999999999999999999999999,0.00008510051840653208772109621169)
2673
(1.089999999999999999999999999,0.0001439985743682589437157612543)
2674
(1.099999999999999999999999999,0.0002290695488414133091768951609)
2675
(1.109999999999999999999999999,0.0003466660264802126738690730543)
2676
(1.119999999999999999999999999,0.0005034933593688608497669215764)
2677
(1.129999999999999999999999999,0.0007064857844317940931934448008)
2678
(1.140000000000000000000000000,0.0009626884321143823233933230639)
2679
(1.149999999999999999999999999,0.001279147875637595843688303618)
2680
(1.160000000000000000000000000,0.001662812929146419109821151587)
2681
(1.170000000000000000000000000,0.002120446668046181927234648455)
2682
(1.180000000000000000000000000,0.002658550078712365184321368998)
2683
(1.190000000000000000000000000,0.003283297316461160337854214003)
2684
(1.199999999999999999999999999,0.004000482234132513685255867935)
2685
(1.210000000000000000000000000,0.004815475617121147853225103055)
2686
(1.220000000000000000000000000,0.005733192406105030758218315150)
2687
(1.230000000000000000000000000,0.006758068091071676347261236389)
2688
(1.240000000000000000000000000,0.007894043407153857660986083815)
2689
(1.250000000000000000000000000,0.009144556444087357389771694576)
2690
(1.260000000000000000000000000,0.01051254128847919638029689031)
2691
(1.270000000000000000000000000,0.01200043234476624205833845910)
2692
(1.280000000000000000000000000,0.01361017352129303922832380879)
2693
(1.290000000000000000000000000,0.01534323151797017615706708754)
2694
(1.300000000000000000000000000,0.01720061250801693469528652546)
2695
(1.310000000000000000000000000,0.01918288156561561567250642741)
2696
(1.320000000000000000000000000,0.02129018425179446896853227640)
2697
(1.330000000000000000000000000,0.02352226983089909660212107827)
2698
(1.340000000000000000000000000,0.02587851564840593018363552850)
2699
(1.350000000000000000000000000,0.02835795225670535909393472584)
2700
(1.360000000000000000000000000,0.03095928892823281608045194248)
2701
(1.370000000000000000000000000,0.03368093924456310958312117680)
2702
(1.380000000000000000000000000,0.03652104649558416920123725803)
2703
(1.390000000000000000000000000,0.03947750866453990643527789483)
2704
(1.400000000000000000000000000,0.04254800281258754219078860285)
2705
(1.410000000000000000000000000,0.04573000871063777661443005433)
2706
(1.420000000000000000000000000,0.04902083159677741183677324186)
2707
(1.430000000000000000000000000,0.05241762396469355776656887777)
2708
(1.440000000000000000000000000,0.05591740631243257550718298054)
2709
(1.450000000000000000000000000,0.05951708680175741661428248386)
2710
(1.460000000000000000000000000,0.06321347979654356889762877102)
2711
(1.470000000000000000000000000,0.06700332326430726815812647156)
2712
(1.480000000000000000000000000,0.07088329503831729842618628027)
2713
(1.490000000000000000000000000,0.07485002794902379595665730811)
2714
(1.500000000000000000000000000,0.07890012384295453171391163767)
2715
\pscircle[linecolor=red](1,0){0.07}
2716
\rput(.3,.4){0.4}
2717
\end{pspicture}
2718
\caption{A curve of rank 5}\label{rank5}
2719
\end{center}
2720
\end{figure}
2721
2722
To see what $\Lambda(E,s)$ looks like in comparison to $L(E,s)$ we graph both the $\Lambda$-series and $L$-series for four curves of different ranks in Figure~\ref{LandLambda}.
2723
%\vspace{.1cm}
2724
\begin{figure}
2725
\begin{center}
2726
\psset{unit=.9in}
2727
\begin{pspicture}(-0.5,-1.5)(3,1.5)
2728
\psgrid[gridcolor=gray]
2729
\rput(1,1.55){$L$-series}
2730
2731
% axes
2732
\psline[linewidth=0.03]{->}(-0.5,0)(3,0)\rput(3.2,0){$x$}
2733
\psline[linewidth=0.03]{->}(0,-1.5)(0,1.5)\rput(0,1.6){$y$}
2734
2735
\psline[linecolor=green]
2736
(0.01999999999999999999999999999,0.01718681534571533496918180083)
2737
(0.03999999999999999999999999999,0.03444418951606351686559314215)
2738
(0.05999999999999999999999999999,0.05174966841014516933157250137)
2739
(0.07999999999999999999999999999,0.06908186280160544819820794758)
2740
(0.09999999999999999999999999999,0.08642042947681662088512749357)
2741
(0.1199999999999999999999999999,0.1037460509857430348738013162)
2742
(0.1399999999999999999999999999,0.1210404141732979373578606002)
2743
(0.1599999999999999999999999999,0.1382861876488565773070042748)
2744
(0.1799999999999999999999999999,0.1554669983417954019661080629)
2745
(0.1999999999999999999999999999,0.1725674072814870642857787885)
2746
(0.2199999999999999999999999999,0.1895728847310976679516816832)
2747
(0.2399999999999999999999999999,0.2064697847958068214452788582)
2748
(0.2599999999999999999999999999,0.2232453196177017933648133222)
2749
(0.2799999999999999999999999999,0.2398875332615821668632056810)
2750
(0.2999999999999999999999999999,0.2563852753882475021395242889)
2751
(0.3199999999999999999999999999,0.2727281748045231994324285658)
2752
(0.3399999999999999999999999999,0.2889066129723036493196936472)
2753
(0.3599999999999999999999999999,0.3049116975522506882266018881)
2754
(0.3799999999999999999999999999,0.3207352360514724657481960818)
2755
(0.3999999999999999999999999999,0.3363697096385155918461264251)
2756
(0.4199999999999999999999999999,0.3518082471833238672339377489)
2757
(0.4399999999999999999999999999,0.3670445995744415805472340298)
2758
(0.4599999999999999999999999999,0.3820731143606595000144047297)
2759
(0.4799999999999999999999999999,0.3968887107595082325517903479)
2760
(0.5000000000000000000000000000,0.4114868550704872899333574729)
2761
(0.5199999999999999999999999999,0.4258635365266695524939245790)
2762
(0.5399999999999999999999999999,0.4400152436143303139560179694)
2763
(0.5599999999999999999999999999,0.4539389408865081287951249319)
2764
(0.5799999999999999999999999999,0.4676320462929016562827460722)
2765
(0.5999999999999999999999999999,0.4810924090452330202713752181)
2766
(0.6199999999999999999999999999,0.4943182880341543604186906474)
2767
(0.6400000000000000000000000000,0.5073083308109308118609439960)
2768
(0.6600000000000000000000000000,0.5200615531444908104381963017)
2769
(0.6800000000000000000000000000,0.5325773191619842190133578282)
2770
(0.6999999999999999999999999999,0.5448553220787213134989496874)
2771
(0.7200000000000000000000000000,0.5568955655212723463175697575)
2772
(0.7400000000000000000000000000,0.5686983454455796124749047414)
2773
(0.7600000000000000000000000000,0.5802642326501632857618388233)
2774
(0.7800000000000000000000000000,0.5915940558828806017375856849)
2775
(0.8000000000000000000000000000,0.6026888855382173064467543532)
2776
(0.8200000000000000000000000000,0.6135500179407429733349907894)
2777
(0.8400000000000000000000000000,0.6241789602091403707947075776)
2778
(0.8600000000000000000000000000,0.6345774156941163456142179008)
2779
(0.8800000000000000000000000000,0.6447472699825107334901933087)
2780
(0.9000000000000000000000000000,0.6546905774590339319229932763)
2781
(0.9200000000000000000000000000,0.6644095484162765437017544146)
2782
(0.9400000000000000000000000000,0.6739065367029397456279240413)
2783
(0.9600000000000000000000000000,0.6831840278996268345536515145)
2784
(0.9800000000000000000000000000,0.6922446280110090786829061099)
2785
(1.000000000000000000000000000,0.7010910526627271305875095398)
2786
(1.020000000000000000000000000,0.7097261167910076565239347125)
2787
(1.040000000000000000000000000,0.7181527248126585597314789803)
2788
(1.060000000000000000000000000,0.7263738612628505121698431233)
2789
(1.080000000000000000000000000,0.7343925818878929763756258124)
2790
(1.100000000000000000000000000,0.7422120051800652358926536927)
2791
(1.120000000000000000000000000,0.7498353043414631145441271942)
2792
(1.140000000000000000000000000,0.7572656996637662173179415238)
2793
(1.160000000000000000000000000,0.7645064513108150382651756859)
2794
(1.180000000000000000000000000,0.7715608524909087202584474283)
2795
(1.200000000000000000000000000,0.7784322230057893750502702092)
2796
(1.220000000000000000000000000,0.7851239031633646210927646777)
2797
(1.240000000000000000000000000,0.7916392480413334895503044088)
2798
(1.260000000000000000000000000,0.7979816220890193749057787873)
2799
(1.280000000000000000000000000,0.8041543940548747184230133856)
2800
(1.300000000000000000000000000,0.8101609322273032205861412028)
2801
(1.320000000000000000000000000,0.8160045999766443432416647033)
2802
(1.340000000000000000000000000,0.8216887515863795884438963795)
2803
(1.360000000000000000000000000,0.8272167283618485716775831989)
2804
(1.380000000000000000000000000,0.8325918550050034164871259150)
2805
(1.400000000000000000000000000,0.8378174362439807853030944270)
2806
(1.420000000000000000000000000,0.8428967537065303466101841214)
2807
(1.440000000000000000000000000,0.8478330630266051943707462961)
2808
(1.460000000000000000000000000,0.8526295911736923225988731765)
2809
(1.480000000000000000000000000,0.8572895339947384594310477314)
2810
(1.500000000000000000000000000,0.8618160539588072213475651896)
2811
(1.520000000000000000000000000,0.8662122780948865917125199484)
2812
(1.540000000000000000000000000,0.8704812961135501778354210042)
2813
(1.560000000000000000000000000,0.8746261587034606587803018492)
2814
(1.580000000000000000000000000,0.8786498759939884811306534807)
2815
(1.600000000000000000000000000,0.8825554161755024438708025645)
2816
(1.620000000000000000000000000,0.8863457042691706572206704566)
2817
(1.640000000000000000000000000,0.8900236210383898490381316271)
2818
(1.660000000000000000000000000,0.8935920020342375723306921073)
2819
(1.680000000000000000000000000,0.8970536367676150414353971752)
2820
(1.700000000000000000000000000,0.9004112680010176487458960935)
2821
(1.720000000000000000000000000,0.9036675911531352945210824755)
2822
(1.740000000000000000000000000,0.9068252538097451518575367410)
2823
(1.760000000000000000000000000,0.9098868553346150832890219622)
2824
(1.780000000000000000000000000,0.9128549465743863610265316789)
2825
(1.800000000000000000000000000,0.9157320296516493934610968850)
2826
(1.820000000000000000000000000,0.9185205578406656349621433238)
2827
(1.840000000000000000000000000,0.9212229355204225952536227502)
2828
(1.860000000000000000000000000,0.9238415181999367396423325301)
2829
(1.880000000000000000000000000,0.9263786126109409805814968830)
2830
(1.900000000000000000000000000,0.9288364768633093283491385600)
2831
(1.920000000000000000000000000,0.9312173206587810412221944903)
2832
(1.940000000000000000000000000,0.9335233055587502620608413631)
2833
(1.960000000000000000000000000,0.9357565453020846368956280639)
2834
(1.980000000000000000000000000,0.9379191061691277880130715350)
2835
(2.000000000000000000000000000,0.9400130073882257814963021214)
2836
(2.020000000000000000000000000,0.9420402215812969242647099595)
2837
(2.040000000000000000000000000,0.9440026752451373987513945643)
2838
(2.060000000000000000000000000,0.9459022492653224568196330320)
2839
(2.080000000000000000000000000,0.9477407794597242214407911249)
2840
(2.100000000000000000000000000,0.9495200571488226676922162200)
2841
(2.120000000000000000000000000,0.9512418297501361649170761909)
2842
(2.140000000000000000000000000,0.9529078013942421580128722224)
2843
(2.160000000000000000000000000,0.9545196335599972528926752815)
2844
(2.180000000000000000000000000,0.9560789457266992599436673580)
2845
(2.200000000000000000000000000,0.9575873160410617553668666970)
2846
(2.220000000000000000000000000,0.9590462819969945632658160834)
2847
(2.240000000000000000000000000,0.9604573411263013642786885482)
2848
(2.260000000000000000000000000,0.9618219516985185251638818473)
2849
(2.280000000000000000000000000,0.9631415334282273459341585496)
2850
(2.300000000000000000000000000,0.9644174681882753663574502411)
2851
(2.320000000000000000000000000,0.9656511007274412924595954006)
2852
(2.340000000000000000000000000,0.9668437393911726272602342846)
2853
(2.360000000000000000000000000,0.9679966568441153497424961906)
2854
(2.380000000000000000000000000,0.9691110907932411132232581627)
2855
(2.400000000000000000000000000,0.9701882447104595595593245443)
2856
(2.420000000000000000000000000,0.9712292885536815988707203537)
2857
(2.440000000000000000000000000,0.9722353594853740144469640266)
2858
(2.460000000000000000000000000,0.9732075625877166466203370990)
2859
(2.480000000000000000000000000,0.9741469715735408134428558314)
2860
(2.500000000000000000000000000,0.9750546294922916639967612353)
2861
(2.520000000000000000000000000,0.9759315494303179541344264483)
2862
(2.540000000000000000000000000,0.9767787152048504042854336739)
2863
(2.560000000000000000000000000,0.9775970820510844623230674031)
2864
(2.580000000000000000000000000,0.9783875773018350666031262487)
2865
(2.600000000000000000000000000,0.9791511010592799979448971263)
2866
(2.620000000000000000000000000,0.9798885268583547347201162044)
2867
(2.640000000000000000000000000,0.9806007023214054899170009741)
2868
(2.660000000000000000000000000,0.9812884498037484179272296000)
2869
(2.680000000000000000000000000,0.9819525670298219340034298152)
2870
(2.700000000000000000000000000,0.9825938277196557902264266074)
2871
(2.720000000000000000000000000,0.9832129822054150949912363593)
2872
(2.740000000000000000000000000,0.9838107580378099422564315965)
2873
(2.760000000000000000000000000,0.9843878605821918230903250927)
2874
(2.780000000000000000000000000,0.9849449736041866135818074281)
2875
(2.800000000000000000000000000,0.9854827598447407553739519456)
2876
(2.820000000000000000000000000,0.9860018615844823505723843165)
2877
(2.840000000000000000000000000,0.9865029011973223614882763875)
2878
(2.860000000000000000000000000,0.9869864816932430148012639218)
2879
(2.880000000000000000000000000,0.9874531872502409338026732446)
2880
(2.900000000000000000000000000,0.9879035837354115333041163899)
2881
(2.920000000000000000000000000,0.9883382192151788788816671071)
2882
(2.940000000000000000000000000,0.9887576244546916021395682079)
2883
(2.960000000000000000000000000,0.9891623134064206408941194847)
2884
(2.980000000000000000000000000,0.9895527836880085994301301208)
2885
(3.000000000000000000000000000,0.9899295170494334587123143226)
2886
2887
\psline[linecolor=blue]
2888
(0.01999999999999999999999999999,-0.007040682645273880424414896490)
2889
(0.03999999999999999999999999999,-0.01384879340673429562897025802)
2890
(0.05999999999999999999999999999,-0.02041151906125685194786353569)
2891
(0.07999999999999999999999999999,-0.02671717089639622967868548846)
2892
(0.09999999999999999999999999999,-0.03275514877468835667228254248)
2893
(0.1199999999999999999999999999,-0.03851590459515947560090356606)
2894
(0.1399999999999999999999999999,-0.04399090531018095579842531960)
2895
(0.1599999999999999999999999999,-0.04917259564426959043634517837)
2896
(0.1799999999999999999999999999,-0.05405436065038169549825383245)
2897
(0.1999999999999999999999999999,-0.05863048822868273853342317371)
2898
(0.2199999999999999999999999999,-0.06289613172268792282287897432)
2899
(0.2399999999999999999999999999,-0.06684727269805707901875306754)
2900
(0.2599999999999999999999999999,-0.07048068400018198510490385939)
2901
(0.2799999999999999999999999999,-0.07379389317801729705702480679)
2902
(0.2999999999999999999999999999,-0.07678514635336806772315486764)
2903
(0.3199999999999999999999999999,-0.07945337260704693367480853604)
2904
(0.3399999999999999999999999999,-0.08179814894594129630869963056)
2905
(0.3599999999999999999999999999,-0.08381966590807343598837635362)
2906
(0.3799999999999999999999999999,-0.08551869385618219352561940146)
2907
(0.3999999999999999999999999999,-0.08689655000419094575245607525)
2908
(0.4199999999999999999999999999,-0.08795506621514009498707863136)
2909
(0.4399999999999999999999999999,-0.08869655760373996378059430584)
2910
(0.4599999999999999999999999999,-0.08912379197162846543221041580)
2911
(0.4799999999999999999999999999,-0.08923996009868375692678919869)
2912
(0.5000000000000000000000000000,-0.08904864690933180697948009138)
2913
(0.5199999999999999999999999999,-0.08855380352868900012719369984)
2914
(0.5399999999999999999999999999,-0.08775972023957721056568979875)
2915
(0.5599999999999999999999999999,-0.08667100034793001370606707268)
2916
(0.5799999999999999999999999999,-0.08529253496086083061378244649)
2917
(0.5999999999999999999999999999,-0.08362947867867400124606939994)
2918
(0.6199999999999999999999999999,-0.08168722619935547704606842491)
2919
(0.6400000000000000000000000000,-0.07947138983156869830977112899)
2920
(0.6600000000000000000000000000,-0.07698777790989125167415357936)
2921
(0.6800000000000000000000000000,-0.07424237410394737051996840888)
2922
(0.6999999999999999999999999999,-0.07124131761120885189838824192)
2923
(0.7200000000000000000000000000,-0.06799088422154145995628282789)
2924
(0.7400000000000000000000000000,-0.06449746824005465646627421466)
2925
(0.7600000000000000000000000000,-0.06076756525345918698261559700)
2926
(0.7800000000000000000000000000,-0.05680775572393965919941740744)
2927
(0.8000000000000000000000000000,-0.05262468939349814449812226300)
2928
(0.8200000000000000000000000000,-0.04822507048081074570172472194)
2929
(0.8400000000000000000000000000,-0.04361564365185310007207290383)
2930
(0.8600000000000000000000000000,-0.03880318074488438664428006043)
2931
(0.8800000000000000000000000000,-0.03379446822982440208845599349)
2932
(0.9000000000000000000000000000,-0.02859629538160683747275401029)
2933
(0.9200000000000000000000000000,-0.02321544314673655974290269603)
2934
(0.9400000000000000000000000000,-0.01765867368201235289112268235)
2935
(0.9600000000000000000000000000,-0.01193272054419242079363246703)
2936
(0.9800000000000000000000000000,-0.006044279509271545062101302395)
2937
(1.000000000000000000000000000,0)
2938
(1.020000000000000000000000000,0.006193522899700657948953702821)
2939
(1.040000000000000000000000000,0.01252975586468649046170633195)
2940
(1.060000000000000000000000000,0.01900223420402991510662878509)
2941
(1.080000000000000000000000000,0.02560456821288883018732881229)
2942
(1.100000000000000000000000000,0.03233044902151853285873105062)
2943
(1.120000000000000000000000000,0.03917365396187151515130066257)
2944
(1.140000000000000000000000000,0.04612805147192782966336624274)
2945
(1.160000000000000000000000000,0.05318760555756510911394608809)
2946
(1.180000000000000000000000000,0.06034637983141611780977478710)
2947
(1.200000000000000000000000000,0.06759854114777629370830779932)
2948
(1.220000000000000000000000000,0.07493836285221727134548049211)
2949
(1.240000000000000000000000000,0.08236022766413781101477395164)
2950
(1.260000000000000000000000000,0.08985863021004364857227589662)
2951
(1.280000000000000000000000000,0.09742817922489508043562107831)
2952
(1.300000000000000000000000000,0.1050635994383979831195623428)
2953
(1.320000000000000000000000000,0.1127597331626426342750000402)
2954
(1.340000000000000000000000000,0.1205115415970171844667067337)
2955
(1.360000000000000000000000000,0.1283141058658407995725197568)
2956
(1.380000000000000000000000000,0.1361626278036770765653872408)
2957
(1.400000000000000000000000000,0.1440524305028029124532985236)
2958
(1.420000000000000000000000000,0.1519789586368230249427400150)
2959
(1.440000000000000000000000000,0.1599377785739371047713213740)
2960
(1.460000000000000000000000000,0.1679245782928863247835066101)
2961
(1.480000000000000000000000000,0.1759351671141297281229851813)
2962
(1.500000000000000000000000000,0.1839654752583298497321186624)
2963
(1.520000000000000000000000000,0.1920115532437616743525970716)
2964
(1.540000000000000000000000000,0.2000695711338004895487135877)
2965
(1.560000000000000000000000000,0.2081358176451930554189518053)
2966
(1.580000000000000000000000000,0.2162066991273734031110841806)
2967
(1.600000000000000000000000000,0.2242787384226500348843475940)
2968
(1.620000000000000000000000000,0.2323485736166658006761623977)
2969
(1.640000000000000000000000000,0.2404129566881156747794000563)
2970
(1.660000000000000000000000000,0.2484687520663013942874635967)
2971
(1.680000000000000000000000000,0.2565129351047057339548403668)
2972
(1.700000000000000000000000000,0.2645425904783833123832212255)
2973
(1.720000000000000000000000000,0.2725549105125894351203245871)
2974
(1.740000000000000000000000000,0.2805471934497037191070854782)
2975
(1.760000000000000000000000000,0.2885168416611512059026883501)
2976
(1.780000000000000000000000000,0.2964613598106804158541689015)
2977
(1.800000000000000000000000000,0.3043783529750253442992429567)
2978
(1.820000000000000000000000000,0.3122655247276567443190829145)
2979
(1.840000000000000000000000000,0.3201206751910171395556735942)
2980
(1.860000000000000000000000000,0.3279416990623337996760249098)
2981
(1.880000000000000000000000000,0.3357265836178143006713637220)
2982
(1.900000000000000000000000000,0.3434734066997501711381013130)
2983
(1.920000000000000000000000000,0.3511803346907853633889956286)
2984
(1.940000000000000000000000000,0.3588456204793477367406288223)
2985
(1.960000000000000000000000000,0.3664676014199932362853021491)
2986
(1.980000000000000000000000000,0.3740446972921738169877932115)
2987
(2.000000000000000000000000000,0.3815754082607112112937104095)
2988
(2.020000000000000000000000000,0.3890583128410391695598633238)
2989
(2.040000000000000000000000000,0.3964920658720666086723007354)
2990
(2.060000000000000000000000000,0.4038753964993129699142493503)
2991
(2.080000000000000000000000000,0.4112071061707747910139988429)
2992
(2.100000000000000000000000000,0.4184860666477988128610003466)
2993
(2.120000000000000000000000000,0.4257112180330616382145949304)
2994
(2.140000000000000000000000000,0.4328815668175888044935202098)
2995
(2.160000000000000000000000000,0.4399961839485868900507079621)
2996
(2.180000000000000000000000000,0.4470542029197107066446098109)
2997
(2.200000000000000000000000000,0.4540548178852435031314417493)
2998
(2.220000000000000000000000000,0.4609972817995311800149907190)
2999
(2.240000000000000000000000000,0.4678809045828815556061141898)
3000
(2.260000000000000000000000000,0.4747050513150164978537264130)
3001
(2.280000000000000000000000000,0.4814691404570480091101373545)
3002
(2.300000000000000000000000000,0.4881726421028388943567929956)
3003
(2.320000000000000000000000000,0.4948150762605042298041400070)
3004
(2.340000000000000000000000000,0.5013960111647112546180296503)
3005
(2.360000000000000000000000000,0.5079150616203423137352231886)
3006
(2.380000000000000000000000000,0.5143718873779978681164907961)
3007
(2.400000000000000000000000000,0.5207661915417341482886693073)
3008
(2.420000000000000000000000000,0.5270977190093525499433942425)
3009
(2.440000000000000000000000000,0.5333662549454851535285664326)
3010
(2.460000000000000000000000000,0.5395716232876525947265009191)
3011
(2.480000000000000000000000000,0.5457136852854067258264170737)
3012
(2.500000000000000000000000000,0.5517923380726109005679500842)
3013
(2.520000000000000000000000000,0.5578075132728551033706124653)
3014
(2.540000000000000000000000000,0.5637591756379513493598187888)
3015
(2.560000000000000000000000000,0.5696473217194066307559695957)
3016
(2.580000000000000000000000000,0.5754719785727260096488165877)
3017
(2.600000000000000000000000000,0.5812332024943570937173917601)
3018
(2.620000000000000000000000000,0.5869310777910489219925825373)
3019
(2.640000000000000000000000000,0.5925657155813630793228431299)
3020
(2.660000000000000000000000000,0.5981372526290425028982153542)
3021
(2.680000000000000000000000000,0.6036458502079137991495723328)
3022
(2.700000000000000000000000000,0.6090916929979718166801361655)
3023
(2.720000000000000000000000000,0.6144749880122705876273751187)
3024
(2.740000000000000000000000000,0.6197959635542224278568417655)
3025
(2.760000000000000000000000000,0.6250548682048868522766910655)
3026
(2.780000000000000000000000000,0.6302519698398128966284810703)
3027
(2.800000000000000000000000000,0.6353875546749823272365581299)
3028
(2.820000000000000000000000000,0.6404619263413869557542058624)
3029
(2.840000000000000000000000000,0.6454754049877607516894869008)
3030
(2.860000000000000000000000000,0.6504283264109765604753116705)
3031
(2.880000000000000000000000000,0.6553210412136078922982997797)
3032
(2.900000000000000000000000000,0.6601539139881483541278100795)
3033
(2.920000000000000000000000000,0.6649273225273747656668244570)
3034
(2.940000000000000000000000000,0.6696416570603347444156881135)
3035
(2.960000000000000000000000000,0.6742973195134354845820148258)
3036
(2.980000000000000000000000000,0.6788947227961075117071180513)
3037
(3.000000000000000000000000000,0.6834342901105152956599508217)
3038
3039
\psline[linecolor=cyan]
3040
(0.01999999999999999999999999999,0.06633205232924863831021756275)
3041
(0.03999999999999999999999999999,0.1238589869993040262250816535)
3042
(0.05999999999999999999999999999,0.1732490974204031901524233845)
3043
(0.07999999999999999999999999999,0.2151452257836850350849844517)
3044
(0.09999999999999999999999999999,0.2501634032128948577781162685)
3045
(0.1199999999999999999999999999,0.2788918829395867975266200164)
3046
(0.1399999999999999999999999999,0.3018905175888066893166510691)
3047
(0.1599999999999999999999999999,0.3196904360975578975797463927)
3048
(0.1799999999999999999999999999,0.3327939799261070683793316003)
3049
(0.1999999999999999999999999999,0.3416748620723204017123107521)
3050
(0.2199999999999999999999999999,0.3467785159727270810670498189)
3051
(0.2399999999999999999999999999,0.3485226046826493404029088088)
3052
(0.2599999999999999999999999999,0.3472976637838162485768392706)
3053
(0.2799999999999999999999999999,0.3434678542840188216894816108)
3054
(0.2999999999999999999999999999,0.3373718043623533659424285446)
3055
(0.3199999999999999999999999999,0.3293235211882361546026495165)
3056
(0.3399999999999999999999999999,0.3196133562153448294365761606)
3057
(0.3599999999999999999999999999,0.3085090093354303941233055163)
3058
(0.3799999999999999999999999999,0.2962565590837378578062775966)
3059
(0.3999999999999999999999999999,0.2830815077294071810755392342)
3060
(0.4199999999999999999999999999,0.2691898315721259382198436763)
3061
(0.4399999999999999999999999999,0.2547690281114520602682614897)
3062
(0.4599999999999999999999999999,0.2399891529681262198170412718)
3063
(0.4799999999999999999999999999,0.2250038405273628209988407423)
3064
(0.5000000000000000000000000000,0.2099513032520556529211768679)
3065
(0.5199999999999999999999999999,0.1949553054880592575852744770)
3066
(0.5399999999999999999999999999,0.1801261083627015878193116810)
3067
(0.5599999999999999999999999999,0.1655613830694359537456005338)
3068
(0.5799999999999999999999999999,0.1513470904435447588671322255)
3069
(0.5999999999999999999999999999,0.1375583252730762514803768786)
3070
(0.6199999999999999999999999999,0.1242601242622731563050750119)
3071
(0.6400000000000000000000000000,0.1115082369777323362159891224)
3072
(0.6600000000000000000000000000,0.09934985946607825450425559998)
3073
(0.6800000000000000000000000000,0.08782433054128655920915869191)
3074
(0.6999999999999999999999999999,0.07696379100480972080761890705)
3075
(0.7200000000000000000000000000,0.06679380628681199812990246179)
3076
(0.7400000000000000000000000000,0.05733395318623945854793812367)
3077
(0.7600000000000000000000000000,0.04859837154492148960574052244)
3078
(0.7800000000000000000000000000,0.04059628181989797667737901674)
3079
(0.8000000000000000000000000000,0.03333246962187091404798015893)
3080
(0.8200000000000000000000000000,0.02680773836899450420488153651)
3081
(0.8400000000000000000000000000,0.02101933126678998630035469414)
3082
(0.8600000000000000000000000000,0.01596132386920678110544201227)
3083
(0.8800000000000000000000000000,0.01162498850493334187192106899)
3084
(0.9000000000000000000000000000,0.007999131868965912642441395577)
3085
(0.9200000000000000000000000000,0.005070407083956478888872728231)
3086
(0.9400000000000000000000000000,0.002823601530591176909816049109)
3087
(0.9600000000000000000000000000,0.001241901732643167124218419275)
3088
(0.9800000000000000000000000000,0.0003071365616956409405041744633)
3089
(1.000000000000000000000000000,0)
3090
(1.020000000000000000000000000,0.0003002546685542481584883817277)
3091
(1.040000000000000000000000000,0.001186917292175145886470395270)
3092
(1.060000000000000000000000000,0.002638427234907597995192163478)
3093
(1.080000000000000000000000000,0.004632799198283402720258234366)
3094
(1.100000000000000000000000000,0.007147761132339355951297819566)
3095
(1.120000000000000000000000000,0.01016087836548349682017904803)
3096
(1.140000000000000000000000000,0.01364966491473845803352122823)
3097
(1.160000000000000000000000000,0.01759168289300856828113180647)
3098
(1.180000000000000000000000000,0.02196463088517071480962718469)
3099
(1.200000000000000000000000000,0.02674642212028410492580900504)
3100
(1.220000000000000000000000000,0.03191525322331037831235999896)
3101
(1.240000000000000000000000000,0.03744966428665069684926975382)
3102
(1.260000000000000000000000000,0.04332859095972094819532206158)
3103
(1.280000000000000000000000000,0.04953140921384744528365415091)
3104
(1.300000000000000000000000000,0.05603797340009158542754001716)
3105
(1.320000000000000000000000000,0.06282864817929507627563652885)
3106
(1.340000000000000000000000000,0.06988433486674700883909261771)
3107
(1.360000000000000000000000000,0.07718649269845979080186917044)
3108
(1.380000000000000000000000000,0.08471715549213489220344650360)
3109
(1.400000000000000000000000000,0.09245894414351856480180224416)
3110
(1.420000000000000000000000000,0.1003950753679962330352313589)
3111
(1.440000000000000000000000000,0.1085093670679450315037515450)
3112
(1.460000000000000000000000000,0.1167862406785404202910496723)
3113
(1.480000000000000000000000000,0.1252107208183704033015454849)
3114
(1.500000000000000000000000000,0.1337684325463184144182903735)
3115
(1.520000000000000000000000000,0.1424455965026967638450474248)
3116
(1.540000000000000000000000000,0.1512290221905055775109462058)
3117
(1.560000000000000000000000000,0.1601060996319128505769381796)
3118
(1.580000000000000000000000000,0.1690647896155523320460112270)
3119
(1.600000000000000000000000000,0.1780936127319682621754391509)
3120
(1.620000000000000000000000000,0.1871816373774489639875751267)
3121
(1.640000000000000000000000000,0.1963184668905336185903967618)
3122
(1.660000000000000000000000000,0.2054942259705965816493012890)
3123
(1.680000000000000000000000000,0.2146995465140597545581439636)
3124
(1.700000000000000000000000000,0.2239255529909046600601038464)
3125
(1.720000000000000000000000000,0.2331638474722015517620026003)
3126
(1.740000000000000000000000000,0.2424064944082936284211799523)
3127
(1.760000000000000000000000000,0.2516460052470219006656846127)
3128
(1.780000000000000000000000000,0.2608753229719034621687023556)
3129
(1.800000000000000000000000000,0.2700878066314372903924979781)
3130
(1.820000000000000000000000000,0.2792772159226632351689109758)
3131
(1.840000000000000000000000000,0.2884376958846991638357231880)
3132
(1.860000000000000000000000000,0.2975637617511876160398143306)
3133
(1.880000000000000000000000000,0.3066502840043577897819130821)
3134
(1.900000000000000000000000000,0.3156924736677139737711561023)
3135
(1.920000000000000000000000000,0.3246858678691621343664513934)
3136
(1.940000000000000000000000000,0.3336263157016484572980805859)
3137
(1.960000000000000000000000000,0.3425099644040751384131523910)
3138
(1.980000000000000000000000000,0.3513332458813491930824106960)
3139
(2.000000000000000000000000000,0.3600928635788807296980040920)
3140
(2.020000000000000000000000000,0.3687857797236508258718531647)
3141
(2.040000000000000000000000000,0.3774092029410902247245086278)
3142
(2.060000000000000000000000000,0.3859605762544244019818914223)
3143
(2.080000000000000000000000000,0.3944375654708254587933895777)
3144
(2.100000000000000000000000000,0.4028380479566454785017768724)
3145
(2.120000000000000000000000000,0.4111601018021694913778216111)
3146
(2.140000000000000000000000000,0.4194019953747003403784799398)
3147
(2.160000000000000000000000000,0.4275621772573550736243327165)
3148
(2.180000000000000000000000000,0.4356392665696967076039051483)
3149
(2.200000000000000000000000000,0.4436320436652311156968651002)
3150
(2.220000000000000000000000000,0.4515394411998522541716632602)
3151
(2.240000000000000000000000000,0.4593605355645067898736147499)
3152
(2.260000000000000000000000000,0.4670945386746592267266241759)
3153
(2.280000000000000000000000000,0.4747407901085595146246071222)
3154
(2.300000000000000000000000000,0.4822987495858363724860086952)
3155
(2.320000000000000000000000000,0.4897679897775514617778094699)
3156
(2.340000000000000000000000000,0.4971481894385431413804738794)
3157
(2.360000000000000000000000000,0.5044391268526555466528667942)
3158
(2.380000000000000000000000000,0.5116406735812815425491346379)
3159
(2.400000000000000000000000000,0.5187527885055396888373777886)
3160
(2.420000000000000000000000000,0.5257755121523492800387175090)
3161
(2.440000000000000000000000000,0.5327089612946578700408722419)
3162
(2.460000000000000000000000000,0.5395533238161070432200454270)
3163
(2.480000000000000000000000000,0.5463088538304895934951863330)
3164
(2.500000000000000000000000000,0.5529758670464501924162602402)
3165
(2.520000000000000000000000000,0.5595547363680079383280405544)
3166
(2.540000000000000000000000000,0.5660458877216291221281075280)
3167
(2.560000000000000000000000000,0.5724497961007487054862753885)
3168
(2.580000000000000000000000000,0.5787669818188262865625489756)
3169
(2.600000000000000000000000000,0.5849980069622239219107713778)
3170
(2.620000000000000000000000000,0.5911434720344065483191188197)
3171
(2.640000000000000000000000000,0.5972040127831886220005883782)
3172
(2.660000000000000000000000000,0.6031802972029809125617581586)
3173
(2.680000000000000000000000000,0.6090730227042273154751206094)
3174
(2.700000000000000000000000000,0.6148829134424614342827547772)
3175
(2.720000000000000000000000000,0.6206107177996550663633346026)
3176
(2.740000000000000000000000000,0.6262572060107743017435208269)
3177
(2.760000000000000000000000000,0.6318231679287025612292137093)
3178
(2.780000000000000000000000000,0.6373094109209325434200891978)
3179
(2.800000000000000000000000000,0.6427167578916698305616933645)
3180
(2.820000000000000000000000000,0.6480460454232290454373140670)
3181
(2.840000000000000000000000000,0.6532981220308382842075416019)
3182
(2.860000000000000000000000000,0.6584738465251984961861606039)
3183
(2.880000000000000000000000000,0.6635740864773710494218020169)
3184
(2.900000000000000000000000000,0.6685997167807884964349752979)
3185
(2.920000000000000000000000000,0.6735516183054001932121618872)
3186
(2.940000000000000000000000000,0.6784306766391756452179075042)
3187
(2.960000000000000000000000000,0.6832377809123940319920346023)
3188
(2.980000000000000000000000000,0.6879738227003481228525514787)
3189
(3.000000000000000000000000000,0.6926396950002846117222293821)
3190
3191
\psline[linecolor=magenta]
3192
(0.01999999999999999999999999999,-0.6941969565423001064579008505)
3193
(0.03999999999999999999999999999,-1.224303060054913888228611661)
3194
(0.05999999999999999999999999999,-1.617036990043072476380405432)
3195
(0.07999999999999999999999999999,-1.895599825260420064276982232)
3196
(0.09999999999999999999999999999,-2.080058567173062689443317490)
3197
(0.1199999999999999999999999999,-2.187697375097209442807822069)
3198
(0.1399999999999999999999999999,-2.233337785910732953117521202)
3199
(0.1599999999999999999999999999,-2.229629434932846597202591093)
3200
(0.1799999999999999999999999999,-2.187312956336142367783540870)
3201
(0.1999999999999999999999999999,-2.115456837327142818018141335)
3202
(0.2199999999999999999999999999,-2.021670043781688359424560791)
3203
(0.2399999999999999999999999999,-1.912292237330765763733304833)
3204
(0.2599999999999999999999999999,-1.792563374456563069666298951)
3205
(0.2799999999999999999999999999,-1.666774424759994937314205875)
3206
(0.2999999999999999999999999999,-1.538400874605764615403251437)
3207
(0.3199999999999999999999999999,-1.410220599075322086377357005)
3208
(0.3399999999999999999999999999,-1.284417593807022454714262376)
3209
(0.3599999999999999999999999999,-1.162672962285572963824604544)
3210
(0.3799999999999999999999999999,-1.046244456166366802310495314)
3211
(0.3999999999999999999999999999,-0.9360357684041337897681360899)
3212
(0.4199999999999999999999999999,-0.8326566829304489846775894104)
3213
(0.4399999999999999999999999999,-0.7364750916176941703400764989)
3214
(0.4599999999999999999999999999,-0.6476618001723166913083888297)
3215
(0.4799999999999999999999999999,-0.5662289600405636110221516461)
3216
(0.5000000000000000000000000000,-0.4920628837884663767162722526)
3217
(0.5199999999999999999999999999,-0.4249519269617209726671399220)
3218
(0.5399999999999999999999999999,-0.3646100502276295382286701887)
3219
(0.5599999999999999999999999999,-0.3106966116287169800938014682)
3220
(0.5799999999999999999999999999,-0.2628328799305563687087769934)
3221
(0.5999999999999999999999999999,-0.2206157061565192332634880189)
3222
(0.6199999999999999999999999999,-0.1836287412558685910948819996)
3223
(0.6400000000000000000000000000,-0.1514515432036038941355468859)
3224
(0.6600000000000000000000000000,-0.1236668764153878637362459807)
3225
(0.6800000000000000000000000000,-0.09986646990233779653602125192)
3226
(0.6999999999999999999999999999,-0.07965546780820809922285492823)
3227
(0.7200000000000000000000000000,-0.06265577658705412018384921392)
3228
(0.7400000000000000000000000000,-0.04850848682044001416921755886)
3229
(0.7600000000000000000000000000,-0.03687552427649674769076936318)
3230
(0.7800000000000000000000000000,-0.02744066402710006863491442098)
3231
(0.8000000000000000000000000000,-0.01991002302575148987446085203)
3232
(0.8200000000000000000000000000,-0.01401213028322172326308234058)
3233
(0.8400000000000000000000000000,-0.009497659451189876915738041398)
3234
(0.8600000000000000000000000000,-0.006138896041403990484653390642)
3235
(0.8800000000000000000000000000,-0.003729000489484266274466262582)
3236
(0.9000000000000000000000000000,-0.002081118652996614152056766787)
3237
(0.9200000000000000000000000000,-0.001027382961363314017406494273)
3238
(0.9400000000000000000000000000,-0.0004178401723634765122120044597)
3239
(0.9600000000000000000000000000,-0.0001193354108090775330798550141)
3240
(0.9800000000000000000000000000,-0.00001437675569523539982970834126)
3241
(1.000000000000000000000000000,0)
3242
(1.020000000000000000000000000,0.00001335076010323709396208854362)
3243
(1.040000000000000000000000000,0.0001029144255272643359860796470)
3244
(1.060000000000000000000000000,0.0003346666611260519448020428584)
3245
(1.080000000000000000000000000,0.0007643291131354103068160924266)
3246
(1.100000000000000000000000000,0.001438329075870453641136902059)
3247
(1.120000000000000000000000000,0.002394706281075908365750437174)
3248
(1.140000000000000000000000000,0.003663964864451018058842766187)
3249
(1.160000000000000000000000000,0.005269869692308771884970831179)
3250
(1.180000000000000000000000000,0.007230187143107076178798332141)
3251
(1.200000000000000000000000000,0.009557371165349786145716519883)
3252
(1.220000000000000000000000000,0.01225919600281414307728846704)
3253
(1.240000000000000000000000000,0.01533933741442756779897108478)
3254
(1.260000000000000000000000000,0.01879790454047723887660114808)
3255
(1.280000000000000000000000000,0.02263192479750214967619458858)
3256
(1.300000000000000000000000000,0.02683578433704244666524467576)
3257
(1.320000000000000000000000000,0.03140162669208463969482120891)
3258
(1.340000000000000000000000000,0.03631971227130751156193470885)
3259
(1.360000000000000000000000000,0.04157874135518094727701674422)
3260
(1.380000000000000000000000000,0.04716614320819906972809900905)
3261
(1.400000000000000000000000000,0.05306833385534747781079476116)
3262
(1.420000000000000000000000000,0.05927094498449120520526767588)
3263
(1.440000000000000000000000000,0.06575902633492268847102645806)
3264
(1.460000000000000000000000000,0.07251722382017324802586003591)
3265
(1.480000000000000000000000000,0.07952993551397631749826337090)
3266
(1.500000000000000000000000000,0.08678144750494967257462341313)
3267
(1.520000000000000000000000000,0.09425605150056515587276714319)
3268
(1.540000000000000000000000000,0.1019381459362585051211530930)
3269
(1.560000000000000000000000000,0.1098123222226609793674436171)
3270
(1.580000000000000000000000000,0.1178634376441322427302334033)
3271
(1.600000000000000000000000000,0.1260766763059780939693254887)
3272
(1.620000000000000000000000000,0.1344375994166443422785904058)
3273
(1.640000000000000000000000000,0.1429321860852856580248726532)
3274
(1.660000000000000000000000000,0.1515468657147449390448351677)
3275
(1.680000000000000000000000000,0.1602685429753365274249350349)
3276
(1.700000000000000000000000000,0.1690846162559851612628322842)
3277
(1.720000000000000000000000000,0.1779829904062208953656092390)
3278
(1.740000000000000000000000000,0.1869520845051851986279852277)
3279
(1.760000000000000000000000000,0.1959808353220252615207756937)
3280
(1.780000000000000000000000000,0.2050586970656590537713607681)
3281
(1.800000000000000000000000000,0.2141756379606673828509888437)
3282
(1.820000000000000000000000000,0.2233221341297726034172697453)
3283
(1.840000000000000000000000000,0.2324891612117428568921548218)
3284
(1.860000000000000000000000000,0.2416681840963529118286738355)
3285
(1.880000000000000000000000000,0.2508511451149711283894650525)
3286
(1.900000000000000000000000000,0.2600304509861604269868339229)
3287
(1.920000000000000000000000000,0.2691989587801167195171382618)
3288
(1.940000000000000000000000000,0.2783499611335646655554892373)
3289
(1.960000000000000000000000000,0.2874771709176397398850025740)
3290
(1.980000000000000000000000000,0.2965747055350691211609569031)
3291
(2.000000000000000000000000000,0.3056370709993943665549003166)
3292
(2.020000000000000000000000000,0.3146591459278403233288422136)
3293
(2.040000000000000000000000000,0.3236361655605233352311955711)
3294
(2.060000000000000000000000000,0.3325637059018157947808852386)
3295
(2.080000000000000000000000000,0.3414376680646639061197502741)
3296
(2.100000000000000000000000000,0.3502542628853236001868016155)
3297
(2.120000000000000000000000000,0.3590099958641800561590573025)
3298
(2.140000000000000000000000000,0.3677016524779047582002084146)
3299
(2.160000000000000000000000000,0.3763262838990468970337686712)
3300
(2.180000000000000000000000000,0.3848811931511300780276354433)
3301
(2.200000000000000000000000000,0.3933639217203174874682380442)
3302
(2.220000000000000000000000000,0.4017722366386150185801703062)
3303
(2.240000000000000000000000000,0.4101041180483072938071798599)
3304
(2.260000000000000000000000000,0.4183577472527792152975220295)
3305
(2.280000000000000000000000000,0.4265314952549865109567053478)
3306
(2.300000000000000000000000000,0.4346239117815307688696096270)
3307
(2.320000000000000000000000000,0.4426337147875023718864778732)
3308
(2.340000000000000000000000000,0.4505597804349194165848217499)
3309
(2.360000000000000000000000000,0.4584011335356586677964917621)
3310
(2.380000000000000000000000000,0.4661569384481976323856783780)
3311
(2.400000000000000000000000000,0.4738264904162215078882895996)
3312
(2.420000000000000000000000000,0.4814092073361560479497626593)
3313
(2.440000000000000000000000000,0.4889046219399322863990937908)
3314
(2.460000000000000000000000000,0.4963123743787402464988274434)
3315
(2.480000000000000000000000000,0.5036322051931582467284077354)
3316
(2.500000000000000000000000000,0.5108639486548272542604836536)
3317
(2.520000000000000000000000000,0.5180075264647537445144708387)
3318
(2.540000000000000000000000000,0.5250629417933500093972620327)
3319
(2.560000000000000000000000000,0.5320302736474403848513098340)
3320
(2.580000000000000000000000000,0.5389096715496600435683203816)
3321
(2.600000000000000000000000000,0.5457013505159362590105184873)
3322
(2.620000000000000000000000000,0.5524055863170584792543541452)
3323
(2.640000000000000000000000000,0.5590227110107027207717825134)
3324
(2.660000000000000000000000000,0.5655531087306685961213413424)
3325
(2.680000000000000000000000000,0.5719972117205058042214602911)
3326
(2.700000000000000000000000000,0.5783554965991442740802876900)
3327
(2.720000000000000000000000000,0.5846284808465924416813272699)
3328
(2.740000000000000000000000000,0.5908167194982262720959959025)
3329
(2.760000000000000000000000000,0.5969208020366532749488519901)
3330
(2.780000000000000000000000000,0.6029413494705972191016122489)
3331
(2.800000000000000000000000000,0.6088790115907074308408645888)
3332
(2.820000000000000000000000000,0.6147344643926488697994771583)
3333
(2.840000000000000000000000000,0.6205084076582734789474214744)
3334
(2.860000000000000000000000000,0.6262015626861078540429399870)
3335
(2.880000000000000000000000000,0.6318146701628156731049755876)
3336
(2.900000000000000000000000000,0.6373484881677044668307021565)
3337
(2.920000000000000000000000000,0.6428037903027443557826033937)
3338
(2.940000000000000000000000000,0.6481813639409507139323591218)
3339
(2.960000000000000000000000000,0.6534820085863529187302520901)
3340
(2.980000000000000000000000000,0.6587065343391271590629544240)
3341
(3.000000000000000000000000000,0.6638557604598125793076126510)
3342
3343
\rput(3.2,1){$E_0$}
3344
</