L(s) = 1 | + (0.669 − 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.309 − 0.951i)15-s + (0.104 − 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.913 − 0.406i)31-s + (−0.669 − 0.743i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.5 − 0.866i)45-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.309 − 0.951i)15-s + (0.104 − 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.913 − 0.406i)31-s + (−0.669 − 0.743i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.5 − 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.335841052 - 2.378474071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.335841052 - 2.378474071i\) |
\(L(1)\) |
\(\approx\) |
\(1.389386941 - 0.7513937231i\) |
\(L(1)\) |
\(\approx\) |
\(1.389386941 - 0.7513937231i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (-0.669 - 0.743i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.978 + 0.207i)T \) |
| 53 | \( 1 + (0.913 + 0.406i)T \) |
| 59 | \( 1 + (0.978 - 0.207i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−18.811776596059733977616631716959, −17.945453857797536658114265611182, −17.37251426675949883253865414841, −16.61492848302539527277531280758, −15.98095406744807360548850747961, −15.11575675721510914364431634040, −14.72749369948693844071670850745, −13.92302467758279489084606176660, −13.491353594216701161763997093664, −12.75727481091883849778555084380, −11.73791049222586238065795228375, −10.873126218159167347824845735, −10.31922209194100070633512956061, −9.80336504807575197352642144438, −8.93726474449652125742064258352, −8.60466092990497669073582920669, −7.40865389206175192811226180876, −6.97051534844279226055114400803, −5.72233165166507523956887659951, −5.3852015188676152915225099747, −4.45844226541932826001970233184, −3.39538101360776848979286158924, −3.10004006433652673634996133093, −2.00205193940519344303174656433, −1.401141605105048023922112408433,
0.66770083087550503438582009667, 1.40371893414752088544014164237, 2.34163468294193862421089210526, 2.83339867313379547162897207802, 3.82691083003133403604560503102, 4.83336068529208246118893201355, 5.6054580272618631132219567348, 6.30776645556735104670212103505, 7.10646076973597136909155214265, 7.71103113155518197615099449446, 8.59390600857570704376947408625, 9.20685824671137645329594270615, 9.709495162420122113995133518079, 10.53810489807981505773481037799, 11.600619481615487002865954601610, 12.20109271963525132269426875615, 12.9108009210634936816627962328, 13.5623570681290179041018906790, 13.98493431592200757835717655509, 14.65925733390090765103943438907, 15.42400720314441067634897872638, 16.39355257410938369365009321269, 16.863708049628298675357784900803, 17.91394871393052361358973915838, 18.11274015771581138488234308665