| L(s) = 1 | − 1.48e17·2-s + 7.29e26·3-s − 1.95e34·4-s + 6.05e39·5-s − 1.08e44·6-s − 4.10e48·7-s + 9.05e51·8-s − 6.86e54·9-s − 8.97e56·10-s + 6.80e59·11-s − 1.42e61·12-s − 2.20e64·13-s + 6.08e65·14-s + 4.41e66·15-s − 5.29e68·16-s + 2.09e70·17-s + 1.01e72·18-s − 1.06e73·19-s − 1.18e74·20-s − 2.99e75·21-s − 1.00e77·22-s − 2.36e77·23-s + 6.60e78·24-s − 2.04e80·25-s + 3.26e81·26-s − 1.03e82·27-s + 8.03e82·28-s + ⋯ |
| L(s) = 1 | − 0.727·2-s + 0.268·3-s − 0.471·4-s + 0.390·5-s − 0.195·6-s − 1.04·7-s + 1.06·8-s − 0.928·9-s − 0.283·10-s + 0.896·11-s − 0.126·12-s − 1.95·13-s + 0.761·14-s + 0.104·15-s − 0.306·16-s + 0.372·17-s + 0.674·18-s − 0.316·19-s − 0.183·20-s − 0.280·21-s − 0.652·22-s − 0.118·23-s + 0.286·24-s − 0.847·25-s + 1.42·26-s − 0.517·27-s + 0.493·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(116-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(58)\) |
\(\approx\) |
\(0.5149035459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5149035459\) |
| \(L(\frac{117}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.48e17T + 4.15e34T^{2} \) |
| 3 | \( 1 - 7.29e26T + 7.39e54T^{2} \) |
| 5 | \( 1 - 6.05e39T + 2.40e80T^{2} \) |
| 7 | \( 1 + 4.10e48T + 1.53e97T^{2} \) |
| 11 | \( 1 - 6.80e59T + 5.75e119T^{2} \) |
| 13 | \( 1 + 2.20e64T + 1.26e128T^{2} \) |
| 17 | \( 1 - 2.09e70T + 3.17e141T^{2} \) |
| 19 | \( 1 + 1.06e73T + 1.13e147T^{2} \) |
| 23 | \( 1 + 2.36e77T + 3.96e156T^{2} \) |
| 29 | \( 1 + 3.92e83T + 1.49e168T^{2} \) |
| 31 | \( 1 - 8.07e85T + 3.21e171T^{2} \) |
| 37 | \( 1 + 4.38e89T + 2.20e180T^{2} \) |
| 41 | \( 1 + 5.70e92T + 2.95e185T^{2} \) |
| 43 | \( 1 + 1.08e94T + 7.06e187T^{2} \) |
| 47 | \( 1 + 2.05e96T + 1.95e192T^{2} \) |
| 53 | \( 1 + 2.60e98T + 1.95e198T^{2} \) |
| 59 | \( 1 + 6.32e101T + 4.44e203T^{2} \) |
| 61 | \( 1 - 5.74e102T + 2.05e205T^{2} \) |
| 67 | \( 1 + 6.22e104T + 9.96e209T^{2} \) |
| 71 | \( 1 - 2.47e106T + 7.84e212T^{2} \) |
| 73 | \( 1 - 1.45e107T + 1.91e214T^{2} \) |
| 79 | \( 1 - 1.16e109T + 1.68e218T^{2} \) |
| 83 | \( 1 - 2.26e110T + 4.94e220T^{2} \) |
| 89 | \( 1 - 2.37e112T + 1.51e224T^{2} \) |
| 97 | \( 1 - 1.57e114T + 3.01e228T^{2} \) |
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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
−13.53990065468160962471643647393, −12.02003519237702010095300105576, −9.973491864417422328539617264579, −9.408103466393664455412542845170, −8.069145521273423323279660229333, −6.56407070981932589080009852442, −4.98364423515761263129592704297, −3.38863784942102540666270583932, −2.01599754294066437223977071382, −0.38263494118340874215799949347,
0.38263494118340874215799949347, 2.01599754294066437223977071382, 3.38863784942102540666270583932, 4.98364423515761263129592704297, 6.56407070981932589080009852442, 8.069145521273423323279660229333, 9.408103466393664455412542845170, 9.973491864417422328539617264579, 12.02003519237702010095300105576, 13.53990065468160962471643647393
