| L(s) = 1 | + 3.33e10·2-s − 1.33e16·3-s + 5.24e20·4-s + 1.09e24·5-s − 4.44e26·6-s − 1.48e29·7-s − 2.20e30·8-s − 6.57e32·9-s + 3.64e34·10-s + 3.06e35·11-s − 6.98e36·12-s + 1.14e38·13-s − 4.96e39·14-s − 1.45e40·15-s − 3.83e41·16-s − 4.51e42·17-s − 2.19e43·18-s + 1.29e44·19-s + 5.73e44·20-s + 1.97e45·21-s + 1.02e46·22-s − 1.62e47·23-s + 2.93e46·24-s − 4.98e47·25-s + 3.83e48·26-s + 1.98e49·27-s − 7.79e49·28-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 0.460·3-s + 0.888·4-s + 0.839·5-s − 0.633·6-s − 1.03·7-s − 0.153·8-s − 0.787·9-s + 1.15·10-s + 0.362·11-s − 0.409·12-s + 0.425·13-s − 1.42·14-s − 0.387·15-s − 1.09·16-s − 1.59·17-s − 1.08·18-s + 0.986·19-s + 0.746·20-s + 0.478·21-s + 0.497·22-s − 1.70·23-s + 0.0707·24-s − 0.294·25-s + 0.584·26-s + 0.824·27-s − 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(70-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+69/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(35)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{71}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.33e10T + 5.90e20T^{2} \) |
| 3 | \( 1 + 1.33e16T + 8.34e32T^{2} \) |
| 5 | \( 1 - 1.09e24T + 1.69e48T^{2} \) |
| 7 | \( 1 + 1.48e29T + 2.05e58T^{2} \) |
| 11 | \( 1 - 3.06e35T + 7.17e71T^{2} \) |
| 13 | \( 1 - 1.14e38T + 7.27e76T^{2} \) |
| 17 | \( 1 + 4.51e42T + 7.96e84T^{2} \) |
| 19 | \( 1 - 1.29e44T + 1.71e88T^{2} \) |
| 23 | \( 1 + 1.62e47T + 9.10e93T^{2} \) |
| 29 | \( 1 + 5.30e50T + 8.04e100T^{2} \) |
| 31 | \( 1 - 1.03e51T + 8.01e102T^{2} \) |
| 37 | \( 1 - 3.96e53T + 1.60e108T^{2} \) |
| 41 | \( 1 - 3.97e55T + 1.91e111T^{2} \) |
| 43 | \( 1 - 2.19e56T + 5.12e112T^{2} \) |
| 47 | \( 1 + 1.05e57T + 2.37e115T^{2} \) |
| 53 | \( 1 + 7.74e58T + 9.44e118T^{2} \) |
| 59 | \( 1 - 4.93e60T + 1.54e122T^{2} \) |
| 61 | \( 1 - 4.27e61T + 1.54e123T^{2} \) |
| 67 | \( 1 - 1.10e63T + 9.98e125T^{2} \) |
| 71 | \( 1 + 1.05e64T + 5.45e127T^{2} \) |
| 73 | \( 1 - 3.43e64T + 3.70e128T^{2} \) |
| 79 | \( 1 + 3.22e65T + 8.63e130T^{2} \) |
| 83 | \( 1 + 1.44e66T + 2.60e132T^{2} \) |
| 89 | \( 1 - 1.81e67T + 3.22e134T^{2} \) |
| 97 | \( 1 - 3.55e68T + 1.22e137T^{2} \) |
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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
−15.96122632515041024550109088627, −14.08661564886554384675425759181, −13.03507879826441797887384240436, −11.47112721253375282028815365013, −9.367970717525828935737881679877, −6.36950965213024748672164192648, −5.64015087618336625205524727929, −3.86294287331884541733476343675, −2.37531976789441885212991925895, 0,
2.37531976789441885212991925895, 3.86294287331884541733476343675, 5.64015087618336625205524727929, 6.36950965213024748672164192648, 9.367970717525828935737881679877, 11.47112721253375282028815365013, 13.03507879826441797887384240436, 14.08661564886554384675425759181, 15.96122632515041024550109088627
