L(s) = 1 | + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (0.642 + 0.766i)13-s + 17-s − i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (0.642 + 0.766i)13-s + 17-s − i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01098550465 + 0.3523981556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01098550465 + 0.3523981556i\) |
\(L(1)\) |
\(\approx\) |
\(0.8737025227 + 0.1031184774i\) |
\(L(1)\) |
\(\approx\) |
\(0.8737025227 + 0.1031184774i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.984 + 0.173i)T \) |
| 11 | \( 1 + (0.984 + 0.173i)T \) |
| 13 | \( 1 + (0.642 + 0.766i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.642 + 0.766i)T \) |
| 31 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.642 - 0.766i)T \) |
| 61 | \( 1 + (-0.342 - 0.939i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.642 + 0.766i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.939 - 0.342i)T \) |
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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
−18.726694933626345219579034662752, −17.904102616418444952029305521493, −16.82128862431418959556488477609, −16.59027195833840132149585238658, −15.694401428926870883177786526721, −15.07824119286318308781708117510, −14.366795719415876714103350823270, −13.69877130560888547470152209680, −12.67519343203184894978068284942, −12.08603590907079991719097846891, −11.65829829952108173896757265735, −10.61516809671405173023124086170, −10.16505560909408919165565986407, −8.98712758838361423404726551786, −8.50102004731227215563591718388, −7.706931084880719801936621319109, −7.11889576862682160294826774142, −6.01310548010654959886652787452, −5.52299268409193273390299485054, −4.3315759613244311808943219423, −3.7135337662651218783410305319, −3.20101021597475758066613374345, −1.82642222116035798910068237129, −0.97588784711404779030218090032, −0.067003325919528941235541465638,
1.07672723559089282534878186311, 1.87250219439015125404321070086, 3.15383393630349874282012404760, 3.762407310802561868195564263013, 4.37387645884797361888342181368, 5.36878567155564833962448915651, 6.27776274147255096087406089328, 7.09581564963331486893959873512, 7.5620284133753672606953526742, 8.57749682580757949031360259749, 9.125136055918669185541270724561, 9.919656448059382658351904061678, 11.01221490078192923794808772686, 11.387594056308431957109981848949, 12.114310912633115389981833577728, 12.72630081841879091261375987885, 13.80128245688741663075525257730, 14.32030048812507060932791015854, 15.08781608967093142616626936102, 15.71019533436064245686115737991, 16.50429758872876250252920859887, 16.92659341971705161916284035282, 17.98660680452562004040187149921, 18.59030726732877733564636482999, 19.27215992034824186311863851747