L(s) = 1 | + (−0.978 − 0.207i)3-s + (−0.587 + 0.809i)5-s + (0.913 + 0.406i)9-s + (0.743 − 0.669i)15-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.587 − 0.809i)31-s + (0.743 − 0.669i)37-s + (0.207 − 0.978i)41-s + (0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯ |
L(s) = 1 | + (−0.978 − 0.207i)3-s + (−0.587 + 0.809i)5-s + (0.913 + 0.406i)9-s + (0.743 − 0.669i)15-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.587 − 0.809i)31-s + (0.743 − 0.669i)37-s + (0.207 − 0.978i)41-s + (0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1030636494 + 0.2517461559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1030636494 + 0.2517461559i\) |
\(L(1)\) |
\(\approx\) |
\(0.6077197093 + 0.02786528186i\) |
\(L(1)\) |
\(\approx\) |
\(0.6077197093 + 0.02786528186i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.743 - 0.669i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.669 - 0.743i)T \) |
| 31 | \( 1 + (-0.587 - 0.809i)T \) |
| 37 | \( 1 + (0.743 - 0.669i)T \) |
| 41 | \( 1 + (0.207 - 0.978i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.207 + 0.978i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.994 + 0.104i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.587 + 0.809i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.406 - 0.913i)T \) |
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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
−18.28587619430751306976833629576, −17.118159187447034862467730943937, −17.06417329688108626289506738029, −16.201556648776795060609152959674, −15.76389616102910689465685758868, −14.91705323336645902199596347668, −14.308041061095523372942652564953, −13.01584428828678669757616855020, −12.65026689284644505806680966200, −12.18718419219938796844634978455, −11.289013210781184886025694091401, −10.76711640063677461387547972997, −10.040223170739120564749421181891, −9.23017254539630069219997143303, −8.41046322713027330076765022514, −7.82457946950640359272309032922, −6.86176073049474957745810946988, −6.14425838778471928371554029707, −5.46037883055607900807787571560, −4.68364143681997339651743554595, −4.080148223298526797742996729772, −3.41819785558027348211737570216, −1.94042314305776390871289029138, −1.2258069763552241911588236000, −0.11934579663862369096269528429,
0.83368595405489502705928835876, 2.09278349877099287549317665068, 2.770330893873517817916816594298, 4.00106751867929745283022129198, 4.33240106316314828973026158506, 5.53546437251604300883104645766, 5.95568531133196587892743755253, 6.946354302855486484520458066154, 7.35697513660034400377392369913, 7.99026270394597718976224720570, 9.18822250799066638106336547234, 9.85299629490999702672721714251, 10.70627096225452287606494675952, 11.26160244654735976293233296346, 11.66987299718728888751306841083, 12.45945955781327003710319793552, 13.20599695511061637118278547727, 13.915630385025170475012113840994, 14.75615987197340689621697012953, 15.53427240432549971097313823111, 15.919130726639049999618906600703, 16.77852510138951875395102092518, 17.4209936715635360853534975612, 18.11571540606592178953050894847, 18.618437552457058815614684494756