L(s) = 1 | + 2.84e9·2-s − 3.79e14·3-s + 5.77e18·4-s − 6.89e20·5-s − 1.07e24·6-s − 9.80e25·7-s + 9.84e27·8-s + 1.68e28·9-s − 1.95e30·10-s − 2.74e31·11-s − 2.19e33·12-s + 2.74e32·13-s − 2.78e35·14-s + 2.61e35·15-s + 1.46e37·16-s − 2.27e37·17-s + 4.80e37·18-s − 1.42e39·19-s − 3.97e39·20-s + 3.72e40·21-s − 7.81e40·22-s + 1.13e41·23-s − 3.73e42·24-s − 3.86e42·25-s + 7.80e41·26-s + 4.18e43·27-s − 5.65e44·28-s + ⋯ |
L(s) = 1 | + 1.87·2-s − 1.06·3-s + 2.50·4-s − 0.331·5-s − 1.99·6-s − 1.64·7-s + 2.81·8-s + 0.132·9-s − 0.619·10-s − 0.474·11-s − 2.66·12-s + 0.0290·13-s − 3.07·14-s + 0.352·15-s + 2.75·16-s − 0.673·17-s + 0.248·18-s − 1.42·19-s − 0.828·20-s + 1.74·21-s − 0.888·22-s + 0.332·23-s − 2.99·24-s − 0.890·25-s + 0.0543·26-s + 0.922·27-s − 4.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(62-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(31)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{63}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 - 2.84e9T + 2.30e18T^{2} \) |
| 3 | \( 1 + 3.79e14T + 1.27e29T^{2} \) |
| 5 | \( 1 + 6.89e20T + 4.33e42T^{2} \) |
| 7 | \( 1 + 9.80e25T + 3.55e51T^{2} \) |
| 11 | \( 1 + 2.74e31T + 3.34e63T^{2} \) |
| 13 | \( 1 - 2.74e32T + 8.92e67T^{2} \) |
| 17 | \( 1 + 2.27e37T + 1.14e75T^{2} \) |
| 19 | \( 1 + 1.42e39T + 1.00e78T^{2} \) |
| 23 | \( 1 - 1.13e41T + 1.16e83T^{2} \) |
| 29 | \( 1 - 2.20e44T + 1.60e89T^{2} \) |
| 31 | \( 1 + 7.87e44T + 9.39e90T^{2} \) |
| 37 | \( 1 - 5.41e47T + 4.57e95T^{2} \) |
| 41 | \( 1 - 3.44e48T + 2.39e98T^{2} \) |
| 43 | \( 1 + 2.45e49T + 4.38e99T^{2} \) |
| 47 | \( 1 - 7.88e50T + 9.95e101T^{2} \) |
| 53 | \( 1 - 4.85e52T + 1.51e105T^{2} \) |
| 59 | \( 1 + 1.37e54T + 1.05e108T^{2} \) |
| 61 | \( 1 - 4.54e53T + 8.03e108T^{2} \) |
| 67 | \( 1 + 8.79e55T + 2.45e111T^{2} \) |
| 71 | \( 1 - 3.27e56T + 8.44e112T^{2} \) |
| 73 | \( 1 + 6.07e56T + 4.59e113T^{2} \) |
| 79 | \( 1 - 1.23e58T + 5.69e115T^{2} \) |
| 83 | \( 1 + 6.61e58T + 1.15e117T^{2} \) |
| 89 | \( 1 + 1.38e59T + 8.18e118T^{2} \) |
| 97 | \( 1 + 3.96e60T + 1.55e121T^{2} \) |
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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
−16.57634577835638455759928324206, −15.42408694129414743263261836839, −13.26463982769579384643463080700, −12.21392220019893156749168549316, −10.75206702783945454322899220034, −6.70471629149157498504648306267, −5.80335275982875559328762633248, −4.21345880439988315458369248470, −2.72237056783497296800658086066, 0,
2.72237056783497296800658086066, 4.21345880439988315458369248470, 5.80335275982875559328762633248, 6.70471629149157498504648306267, 10.75206702783945454322899220034, 12.21392220019893156749168549316, 13.26463982769579384643463080700, 15.42408694129414743263261836839, 16.57634577835638455759928324206