L(s) = 1 | + (−0.207 + 0.978i)3-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.587 − 0.809i)13-s + (−0.743 + 0.669i)17-s + (0.978 − 0.207i)19-s + (−0.994 − 0.104i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (0.207 + 0.978i)33-s + (0.406 − 0.913i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)3-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.587 − 0.809i)13-s + (−0.743 + 0.669i)17-s + (0.978 − 0.207i)19-s + (−0.994 − 0.104i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (0.207 + 0.978i)33-s + (0.406 − 0.913i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8071651376 - 0.4109950018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8071651376 - 0.4109950018i\) |
\(L(1)\) |
\(\approx\) |
\(0.8707452268 + 0.09857973252i\) |
\(L(1)\) |
\(\approx\) |
\(0.8707452268 + 0.09857973252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.207 + 0.978i)T \) |
| 11 | \( 1 + (0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (-0.743 + 0.669i)T \) |
| 19 | \( 1 + (0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.994 - 0.104i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.406 - 0.913i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.743 - 0.669i)T \) |
| 53 | \( 1 + (0.207 - 0.978i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (0.104 - 0.994i)T \) |
| 67 | \( 1 + (-0.743 + 0.669i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.406 - 0.913i)T \) |
| 79 | \( 1 + (0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.104 - 0.994i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.72810901493441786185366540139, −19.89502268250473430982954870460, −19.52613585955756601516450358560, −18.53978223416541628079765957353, −17.96335356574275182412996314749, −17.237130643902678756155376244157, −16.54096058812112312583044666664, −15.69024353904789926080725377185, −14.534332940311990061062385946976, −14.02881157549690122131723347435, −13.337030208915312485497095707433, −12.29588158315770898232904028305, −11.80034578551430494504261593673, −11.24280969443350506609959083882, −9.92653774823875801905476416811, −9.26749898863917343790268889849, −8.31655657595121299003612723035, −7.40690777431409244920969947815, −6.80064061598177444710492970303, −6.084965152902547145086075422088, −5.03029469573524293575024031651, −4.16383617141239134534334420146, −2.92369471513475027177937403121, −1.99269843244834585442801020003, −1.18022463636111871428826458256,
0.37258143768955905927793721520, 1.87857059729130690902495138268, 3.126176451334258254233853799377, 3.78813202893647197550119241902, 4.69320125508062855494965461833, 5.555505861843669977870872581596, 6.26517190412015233373296637888, 7.32551196168219575689211657433, 8.404534782990993520397427663037, 9.07036761284101442526355616647, 9.90188767029347534475021814804, 10.55196556282915567254850572984, 11.43064631094991881895379656575, 12.018605649417696101739578299133, 13.03347407617353903847986888214, 14.01572380162464621280769706721, 14.738218165351114128866622451427, 15.3278824601404559041239631840, 16.24727160063897483288831751839, 16.73842591411492558320980093716, 17.65718949025346366443479235818, 18.14745828331826286681989158869, 19.541139718802285033560415662555, 19.949112365641460949520776566717, 20.59469277420242557104545209157