| L(s) = 1 | + 2.20e11·2-s + 1.08e18·3-s + 1.09e22·4-s − 2.30e25·5-s + 2.40e29·6-s + 1.96e31·7-s − 5.92e33·8-s + 5.77e35·9-s − 5.09e36·10-s + 9.40e38·11-s + 1.19e40·12-s + 1.14e42·13-s + 4.32e42·14-s − 2.51e43·15-s − 1.72e45·16-s + 1.79e46·17-s + 1.27e47·18-s + 1.43e48·19-s − 2.52e47·20-s + 2.13e49·21-s + 2.07e50·22-s − 1.23e51·23-s − 6.44e51·24-s − 2.59e52·25-s + 2.52e53·26-s − 3.37e52·27-s + 2.14e53·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 1.39·3-s + 0.290·4-s − 0.141·5-s + 1.58·6-s + 0.399·7-s − 0.806·8-s + 0.948·9-s − 0.161·10-s + 0.833·11-s + 0.405·12-s + 1.92·13-s + 0.453·14-s − 0.197·15-s − 1.20·16-s + 1.29·17-s + 1.07·18-s + 1.60·19-s − 0.0411·20-s + 0.557·21-s + 0.947·22-s − 1.06·23-s − 1.12·24-s − 0.979·25-s + 2.18·26-s − 0.0712·27-s + 0.115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(6.075204432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.075204432\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.20e11T + 3.77e22T^{2} \) |
| 3 | \( 1 - 1.08e18T + 6.08e35T^{2} \) |
| 5 | \( 1 + 2.30e25T + 2.64e52T^{2} \) |
| 7 | \( 1 - 1.96e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 9.40e38T + 1.27e78T^{2} \) |
| 13 | \( 1 - 1.14e42T + 3.51e83T^{2} \) |
| 17 | \( 1 - 1.79e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.43e48T + 8.06e95T^{2} \) |
| 23 | \( 1 + 1.23e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 2.48e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 1.95e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 2.65e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.61e59T + 9.09e120T^{2} \) |
| 43 | \( 1 - 8.98e59T + 3.23e122T^{2} \) |
| 47 | \( 1 - 2.48e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 3.59e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 4.00e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 1.59e67T + 7.93e133T^{2} \) |
| 67 | \( 1 - 1.61e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 2.63e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 2.52e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 9.29e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 7.84e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 2.05e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 4.10e74T + 1.01e149T^{2} \) |
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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
−15.67286189928881532815788560204, −14.23435972480361898972743107505, −13.66522268166784773511616252889, −11.82830003468771494801268332058, −9.303343689316899396095369159019, −7.981619235511270078094357245344, −5.83359990030070811993798679782, −3.91373573657624779746893925842, −3.26020102931758188694564734759, −1.44169842225286580724395584792,
1.44169842225286580724395584792, 3.26020102931758188694564734759, 3.91373573657624779746893925842, 5.83359990030070811993798679782, 7.981619235511270078094357245344, 9.303343689316899396095369159019, 11.82830003468771494801268332058, 13.66522268166784773511616252889, 14.23435972480361898972743107505, 15.67286189928881532815788560204
