| L(s) = 1 | + 8.60e15·2-s − 5.90e26·3-s − 2.52e33·4-s + 1.16e38·5-s − 5.07e42·6-s + 5.43e46·7-s − 4.40e49·8-s + 2.57e53·9-s + 1.00e54·10-s − 6.09e57·11-s + 1.48e60·12-s − 8.19e61·13-s + 4.67e62·14-s − 6.89e64·15-s + 6.16e66·16-s − 1.24e68·17-s + 2.21e69·18-s − 1.04e71·19-s − 2.94e71·20-s − 3.20e73·21-s − 5.24e73·22-s − 1.62e75·23-s + 2.59e76·24-s − 3.71e77·25-s − 7.04e77·26-s − 9.78e79·27-s − 1.36e80·28-s + ⋯ |
| L(s) = 1 | + 0.168·2-s − 1.95·3-s − 0.971·4-s + 0.188·5-s − 0.329·6-s + 0.679·7-s − 0.332·8-s + 2.81·9-s + 0.0317·10-s − 0.971·11-s + 1.89·12-s − 1.22·13-s + 0.114·14-s − 0.367·15-s + 0.915·16-s − 0.636·17-s + 0.475·18-s − 1.12·19-s − 0.182·20-s − 1.32·21-s − 0.164·22-s − 0.431·23-s + 0.650·24-s − 0.964·25-s − 0.207·26-s − 3.54·27-s − 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(112-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(56)\) |
\(\approx\) |
\(0.05524660876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05524660876\) |
| \(L(\frac{113}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 8.60e15T + 2.59e33T^{2} \) |
| 3 | \( 1 + 5.90e26T + 9.12e52T^{2} \) |
| 5 | \( 1 - 1.16e38T + 3.85e77T^{2} \) |
| 7 | \( 1 - 5.43e46T + 6.39e93T^{2} \) |
| 11 | \( 1 + 6.09e57T + 3.93e115T^{2} \) |
| 13 | \( 1 + 8.19e61T + 4.44e123T^{2} \) |
| 17 | \( 1 + 1.24e68T + 3.80e136T^{2} \) |
| 19 | \( 1 + 1.04e71T + 8.74e141T^{2} \) |
| 23 | \( 1 + 1.62e75T + 1.41e151T^{2} \) |
| 29 | \( 1 + 2.09e81T + 2.11e162T^{2} \) |
| 31 | \( 1 - 2.01e82T + 3.47e165T^{2} \) |
| 37 | \( 1 - 7.20e86T + 1.17e174T^{2} \) |
| 41 | \( 1 + 2.11e89T + 1.04e179T^{2} \) |
| 43 | \( 1 + 3.42e90T + 2.06e181T^{2} \) |
| 47 | \( 1 - 6.21e92T + 4.00e185T^{2} \) |
| 53 | \( 1 + 5.97e95T + 2.48e191T^{2} \) |
| 59 | \( 1 + 3.30e98T + 3.66e196T^{2} \) |
| 61 | \( 1 + 5.88e98T + 1.48e198T^{2} \) |
| 67 | \( 1 + 2.94e101T + 4.94e202T^{2} \) |
| 71 | \( 1 - 4.24e102T + 3.08e205T^{2} \) |
| 73 | \( 1 + 6.06e102T + 6.74e206T^{2} \) |
| 79 | \( 1 + 7.31e104T + 4.33e210T^{2} \) |
| 83 | \( 1 - 4.60e105T + 1.04e213T^{2} \) |
| 89 | \( 1 - 7.96e107T + 2.41e216T^{2} \) |
| 97 | \( 1 + 1.72e110T + 3.40e220T^{2} \) |
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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
−13.23730808810311289705386683017, −12.20484922297496371264945485628, −10.87329924008863217096218107791, −9.762375887595205237064392818013, −7.67772158602629151786510101601, −6.03525391694381343597848397554, −5.03370359526233425892182027025, −4.36222286036694157487013756539, −1.79384236457169448404621674905, −0.12786647941918826008462107036,
0.12786647941918826008462107036, 1.79384236457169448404621674905, 4.36222286036694157487013756539, 5.03370359526233425892182027025, 6.03525391694381343597848397554, 7.67772158602629151786510101601, 9.762375887595205237064392818013, 10.87329924008863217096218107791, 12.20484922297496371264945485628, 13.23730808810311289705386683017
