| L(s) = 1 | − 1.57e17·2-s − 1.34e27·3-s + 1.45e34·4-s + 3.80e39·5-s + 2.12e44·6-s + 5.08e47·7-s − 6.50e50·8-s + 9.95e53·9-s − 5.99e56·10-s + 1.24e59·11-s − 1.95e61·12-s − 8.80e62·13-s − 8.01e64·14-s − 5.12e66·15-s − 4.80e67·16-s + 2.20e69·17-s − 1.57e71·18-s − 2.31e72·19-s + 5.51e73·20-s − 6.85e74·21-s − 1.96e76·22-s + 3.34e76·23-s + 8.76e77·24-s + 4.81e78·25-s + 1.38e80·26-s − 2.34e80·27-s + 7.37e81·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 1.48·3-s + 1.39·4-s + 1.22·5-s + 2.30·6-s + 0.907·7-s − 0.614·8-s + 1.21·9-s − 1.89·10-s + 1.80·11-s − 2.07·12-s − 1.01·13-s − 1.40·14-s − 1.82·15-s − 0.445·16-s + 0.666·17-s − 1.87·18-s − 1.30·19-s + 1.71·20-s − 1.34·21-s − 2.79·22-s + 0.386·23-s + 0.913·24-s + 0.500·25-s + 1.57·26-s − 0.314·27-s + 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(114-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+56.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(57)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{115}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.57e17T + 1.03e34T^{2} \) |
| 3 | \( 1 + 1.34e27T + 8.21e53T^{2} \) |
| 5 | \( 1 - 3.80e39T + 9.62e78T^{2} \) |
| 7 | \( 1 - 5.08e47T + 3.13e95T^{2} \) |
| 11 | \( 1 - 1.24e59T + 4.75e117T^{2} \) |
| 13 | \( 1 + 8.80e62T + 7.50e125T^{2} \) |
| 17 | \( 1 - 2.20e69T + 1.09e139T^{2} \) |
| 19 | \( 1 + 2.31e72T + 3.15e144T^{2} \) |
| 23 | \( 1 - 3.34e76T + 7.50e153T^{2} \) |
| 29 | \( 1 - 2.01e82T + 1.78e165T^{2} \) |
| 31 | \( 1 + 1.88e84T + 3.34e168T^{2} \) |
| 37 | \( 1 + 7.16e88T + 1.60e177T^{2} \) |
| 41 | \( 1 + 1.69e90T + 1.75e182T^{2} \) |
| 43 | \( 1 - 2.52e92T + 3.81e184T^{2} \) |
| 47 | \( 1 + 3.45e94T + 8.85e188T^{2} \) |
| 53 | \( 1 + 1.85e96T + 6.96e194T^{2} \) |
| 59 | \( 1 - 6.44e98T + 1.27e200T^{2} \) |
| 61 | \( 1 + 5.89e100T + 5.52e201T^{2} \) |
| 67 | \( 1 + 2.40e103T + 2.22e206T^{2} \) |
| 71 | \( 1 + 2.63e104T + 1.55e209T^{2} \) |
| 73 | \( 1 - 1.44e105T + 3.59e210T^{2} \) |
| 79 | \( 1 + 1.48e107T + 2.70e214T^{2} \) |
| 83 | \( 1 - 2.41e107T + 7.17e216T^{2} \) |
| 89 | \( 1 + 8.88e109T + 1.91e220T^{2} \) |
| 97 | \( 1 - 1.58e111T + 3.20e224T^{2} \) |
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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
−12.15865145973597457845832371632, −11.02094846306589834789109094291, −10.01561974117250844255408573912, −8.892016013379651163504057746537, −7.07483198894407636765333982877, −6.06578527801408555472778148126, −4.73315407642471222211176170622, −1.85282784867443322936507326404, −1.25469108106140853782105008951, 0,
1.25469108106140853782105008951, 1.85282784867443322936507326404, 4.73315407642471222211176170622, 6.06578527801408555472778148126, 7.07483198894407636765333982877, 8.892016013379651163504057746537, 10.01561974117250844255408573912, 11.02094846306589834789109094291, 12.15865145973597457845832371632
