| L(s) = 1 | + 3.43e10·2-s + 1.90e16·3-s + 1.18e21·4-s − 4.48e24·5-s + 6.55e26·6-s + 7.01e29·7-s + 4.05e31·8-s − 7.14e33·9-s − 1.53e35·10-s + 1.39e36·11-s + 2.25e37·12-s + 1.53e39·13-s + 2.40e40·14-s − 8.54e40·15-s + 1.39e42·16-s − 5.62e43·17-s − 2.45e44·18-s + 1.79e45·19-s − 5.29e45·20-s + 1.33e46·21-s + 4.79e46·22-s − 1.05e48·23-s + 7.73e47·24-s − 2.22e49·25-s + 5.28e49·26-s − 2.79e50·27-s + 8.27e50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.220·3-s + 0.5·4-s − 0.688·5-s + 0.155·6-s + 0.699·7-s + 0.353·8-s − 0.951·9-s − 0.486·10-s + 0.149·11-s + 0.110·12-s + 0.438·13-s + 0.494·14-s − 0.151·15-s + 0.250·16-s − 1.17·17-s − 0.672·18-s + 0.723·19-s − 0.344·20-s + 0.154·21-s + 0.105·22-s − 0.482·23-s + 0.0778·24-s − 0.525·25-s + 0.310·26-s − 0.429·27-s + 0.349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3.43e10T \) |
| good | 3 | \( 1 - 1.90e16T + 7.50e33T^{2} \) |
| 5 | \( 1 + 4.48e24T + 4.23e49T^{2} \) |
| 7 | \( 1 - 7.01e29T + 1.00e60T^{2} \) |
| 11 | \( 1 - 1.39e36T + 8.68e73T^{2} \) |
| 13 | \( 1 - 1.53e39T + 1.23e79T^{2} \) |
| 17 | \( 1 + 5.62e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 1.79e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 1.05e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 2.91e51T + 6.76e103T^{2} \) |
| 31 | \( 1 + 1.29e53T + 7.70e105T^{2} \) |
| 37 | \( 1 + 4.26e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 1.24e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.00e58T + 9.46e115T^{2} \) |
| 47 | \( 1 + 1.24e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 1.01e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 9.41e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 9.88e62T + 5.73e126T^{2} \) |
| 67 | \( 1 + 9.00e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 8.82e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 2.67e66T + 1.97e132T^{2} \) |
| 79 | \( 1 + 8.08e66T + 5.38e134T^{2} \) |
| 83 | \( 1 - 4.29e67T + 1.79e136T^{2} \) |
| 89 | \( 1 - 2.26e69T + 2.55e138T^{2} \) |
| 97 | \( 1 + 6.06e70T + 1.15e141T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.63361793284463923920454924653, −11.90234891982314558925359173176, −11.00039105056698429365953760426, −8.770700953808519207493723671725, −7.50027401100890012328826874038, −5.83784395875611309761111305590, −4.43379002627418169245006057609, −3.22909719523065032916102426987, −1.78842185483274139114732301462, 0,
1.78842185483274139114732301462, 3.22909719523065032916102426987, 4.43379002627418169245006057609, 5.83784395875611309761111305590, 7.50027401100890012328826874038, 8.770700953808519207493723671725, 11.00039105056698429365953760426, 11.90234891982314558925359173176, 13.63361793284463923920454924653
