| L(s) = 1 | − 1.07e9·2-s − 3.51e14·3-s + 1.15e18·4-s − 2.66e21·5-s + 3.77e23·6-s − 4.09e24·7-s − 1.23e27·8-s − 3.69e27·9-s + 2.86e30·10-s + 5.96e31·11-s − 4.05e32·12-s − 1.23e34·13-s + 4.39e33·14-s + 9.37e35·15-s + 1.32e36·16-s + 3.56e37·17-s + 3.96e36·18-s + 1.17e39·19-s − 3.07e39·20-s + 1.43e39·21-s − 6.40e40·22-s + 2.28e41·23-s + 4.35e41·24-s + 2.77e42·25-s + 1.32e43·26-s + 4.59e43·27-s − 4.72e42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.985·3-s + 0.5·4-s − 1.28·5-s + 0.696·6-s − 0.0686·7-s − 0.353·8-s − 0.0290·9-s + 0.905·10-s + 1.03·11-s − 0.492·12-s − 1.31·13-s + 0.0485·14-s + 1.26·15-s + 0.250·16-s + 1.05·17-s + 0.0205·18-s + 1.17·19-s − 0.640·20-s + 0.0676·21-s − 0.728·22-s + 0.669·23-s + 0.348·24-s + 0.640·25-s + 0.926·26-s + 1.01·27-s − 0.0343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(62-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+61/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(31)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{63}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.07e9T \) |
| good | 3 | \( 1 + 3.51e14T + 1.27e29T^{2} \) |
| 5 | \( 1 + 2.66e21T + 4.33e42T^{2} \) |
| 7 | \( 1 + 4.09e24T + 3.55e51T^{2} \) |
| 11 | \( 1 - 5.96e31T + 3.34e63T^{2} \) |
| 13 | \( 1 + 1.23e34T + 8.92e67T^{2} \) |
| 17 | \( 1 - 3.56e37T + 1.14e75T^{2} \) |
| 19 | \( 1 - 1.17e39T + 1.00e78T^{2} \) |
| 23 | \( 1 - 2.28e41T + 1.16e83T^{2} \) |
| 29 | \( 1 + 2.93e44T + 1.60e89T^{2} \) |
| 31 | \( 1 - 5.26e45T + 9.39e90T^{2} \) |
| 37 | \( 1 - 6.78e47T + 4.57e95T^{2} \) |
| 41 | \( 1 + 2.31e49T + 2.39e98T^{2} \) |
| 43 | \( 1 + 1.05e50T + 4.38e99T^{2} \) |
| 47 | \( 1 + 3.23e50T + 9.95e101T^{2} \) |
| 53 | \( 1 + 2.98e52T + 1.51e105T^{2} \) |
| 59 | \( 1 - 5.26e53T + 1.05e108T^{2} \) |
| 61 | \( 1 - 8.65e53T + 8.03e108T^{2} \) |
| 67 | \( 1 - 6.01e55T + 2.45e111T^{2} \) |
| 71 | \( 1 - 1.04e56T + 8.44e112T^{2} \) |
| 73 | \( 1 + 9.24e56T + 4.59e113T^{2} \) |
| 79 | \( 1 - 7.21e56T + 5.69e115T^{2} \) |
| 83 | \( 1 - 5.19e58T + 1.15e117T^{2} \) |
| 89 | \( 1 - 3.90e59T + 8.18e118T^{2} \) |
| 97 | \( 1 + 6.73e60T + 1.55e121T^{2} \) |
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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
−14.82123260203159160407233486182, −11.99542366409733541458783606470, −11.55686676378844034081431118762, −9.792468685152832206394881552944, −7.982992004646935464520457710135, −6.73571185836178603086825445471, −5.00934390563493473586795161029, −3.24963599775784645019296995117, −1.02881456529642829131452943346, 0,
1.02881456529642829131452943346, 3.24963599775784645019296995117, 5.00934390563493473586795161029, 6.73571185836178603086825445471, 7.982992004646935464520457710135, 9.792468685152832206394881552944, 11.55686676378844034081431118762, 11.99542366409733541458783606470, 14.82123260203159160407233486182
