L(s) = 1 | + (0.947 + 0.319i)2-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (0.907 − 0.419i)7-s + (0.561 + 0.827i)8-s + (0.647 − 0.762i)10-s + (−0.267 + 0.963i)11-s + (0.468 − 0.883i)13-s + (0.994 − 0.108i)14-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + (0.976 − 0.214i)19-s + (0.856 − 0.515i)20-s + (−0.561 + 0.827i)22-s + (0.161 − 0.986i)23-s + ⋯ |
L(s) = 1 | + (0.947 + 0.319i)2-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (0.907 − 0.419i)7-s + (0.561 + 0.827i)8-s + (0.647 − 0.762i)10-s + (−0.267 + 0.963i)11-s + (0.468 − 0.883i)13-s + (0.994 − 0.108i)14-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + (0.976 − 0.214i)19-s + (0.856 − 0.515i)20-s + (−0.561 + 0.827i)22-s + (0.161 − 0.986i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.041878355 - 0.07943158016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.041878355 - 0.07943158016i\) |
\(L(1)\) |
\(\approx\) |
\(2.249626053 + 0.08020651625i\) |
\(L(1)\) |
\(\approx\) |
\(2.249626053 + 0.08020651625i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 \) |
good | 2 | \( 1 + (0.947 + 0.319i)T \) |
| 5 | \( 1 + (0.370 - 0.928i)T \) |
| 7 | \( 1 + (0.907 - 0.419i)T \) |
| 11 | \( 1 + (-0.267 + 0.963i)T \) |
| 13 | \( 1 + (0.468 - 0.883i)T \) |
| 17 | \( 1 + (-0.907 - 0.419i)T \) |
| 19 | \( 1 + (0.976 - 0.214i)T \) |
| 23 | \( 1 + (0.161 - 0.986i)T \) |
| 29 | \( 1 + (0.947 - 0.319i)T \) |
| 31 | \( 1 + (0.976 + 0.214i)T \) |
| 37 | \( 1 + (-0.561 + 0.827i)T \) |
| 41 | \( 1 + (0.161 + 0.986i)T \) |
| 43 | \( 1 + (0.267 + 0.963i)T \) |
| 47 | \( 1 + (0.370 + 0.928i)T \) |
| 53 | \( 1 + (-0.647 - 0.762i)T \) |
| 61 | \( 1 + (-0.947 - 0.319i)T \) |
| 67 | \( 1 + (-0.561 - 0.827i)T \) |
| 71 | \( 1 + (0.370 + 0.928i)T \) |
| 73 | \( 1 + (-0.994 + 0.108i)T \) |
| 79 | \( 1 + (-0.856 + 0.515i)T \) |
| 83 | \( 1 + (-0.0541 - 0.998i)T \) |
| 89 | \( 1 + (0.947 - 0.319i)T \) |
| 97 | \( 1 + (-0.994 - 0.108i)T \) |
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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
−27.11586175573876261819811114623, −26.17091033557887057148601216648, −24.97947390482582644687702385315, −24.15424003161699526486370979256, −23.30836395586583418903288078429, −22.10152224006672102559858964889, −21.54606838718849615826684478542, −20.79276913014616820271760125420, −19.37869680879483893121899932060, −18.60063681938142404870239201596, −17.52501307925356800952071729255, −15.95878372870756345696481077581, −15.1725009966147288799528891151, −13.90236422245836081736055175791, −13.75360924345207279482884594832, −11.99730764651778329964476364139, −11.219440721710407655286945581511, −10.46732952742699699906086014530, −8.937689248109994795798251411241, −7.396250163369128582170182297004, −6.228433072346743155570415088024, −5.34106997268136312500305109020, −3.91999419848437627308739922643, −2.71280188035853690148391564691, −1.56231749133526072479165510921,
1.27229756471746695034714914184, 2.71875023294109448600892487902, 4.55705974987647105350417032648, 4.90521336629031936340499762346, 6.308430647377314810533507063, 7.63460464264306582131715790771, 8.51331950529000300016559770360, 10.12211925994029132628418334747, 11.358885264437285839435167547422, 12.414257619277409741158688932161, 13.30970104275094517199310122835, 14.12625025640346143077035220459, 15.33587794829899005382569687212, 16.120014520492120472086736828560, 17.39882826220837857996043893654, 17.87686116737003065783336367242, 20.06092626594723102835084510106, 20.49274390652598382325781597579, 21.25319589564568977569485916440, 22.52379447446235678985175714046, 23.30179846358678117977978375964, 24.44642153100828093755966721327, 24.803404737106376075682053321505, 25.9337351346366740468251785569, 27.076277324099843802353566648365