| L(s) = 1 | + 5.49e11·2-s − 3.97e18·3-s + 3.02e23·4-s + 4.89e27·5-s − 2.18e30·6-s − 4.89e32·7-s + 1.66e35·8-s − 3.34e37·9-s + 2.69e39·10-s − 5.46e40·11-s − 1.20e42·12-s + 1.00e44·13-s − 2.68e44·14-s − 1.94e46·15-s + 9.13e46·16-s − 6.88e48·17-s − 1.83e49·18-s − 1.77e50·19-s + 1.47e51·20-s + 1.94e51·21-s − 3.00e52·22-s − 4.07e53·23-s − 6.60e53·24-s + 7.41e54·25-s + 5.52e55·26-s + 3.29e56·27-s − 1.47e56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.566·3-s + 0.5·4-s + 1.20·5-s − 0.400·6-s − 0.203·7-s + 0.353·8-s − 0.678·9-s + 0.850·10-s − 0.400·11-s − 0.283·12-s + 1.00·13-s − 0.143·14-s − 0.682·15-s + 0.250·16-s − 1.71·17-s − 0.479·18-s − 0.548·19-s + 0.601·20-s + 0.115·21-s − 0.283·22-s − 0.663·23-s − 0.200·24-s + 0.448·25-s + 0.709·26-s + 0.951·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+79/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(40)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.49e11T \) |
| good | 3 | \( 1 + 3.97e18T + 4.92e37T^{2} \) |
| 5 | \( 1 - 4.89e27T + 1.65e55T^{2} \) |
| 7 | \( 1 + 4.89e32T + 5.79e66T^{2} \) |
| 11 | \( 1 + 5.46e40T + 1.86e82T^{2} \) |
| 13 | \( 1 - 1.00e44T + 1.00e88T^{2} \) |
| 17 | \( 1 + 6.88e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 1.77e50T + 1.05e101T^{2} \) |
| 23 | \( 1 + 4.07e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 6.28e56T + 3.38e115T^{2} \) |
| 31 | \( 1 - 8.31e58T + 6.57e117T^{2} \) |
| 37 | \( 1 - 3.89e61T + 7.72e123T^{2} \) |
| 41 | \( 1 + 2.06e63T + 2.56e127T^{2} \) |
| 43 | \( 1 + 6.18e64T + 1.10e129T^{2} \) |
| 47 | \( 1 - 1.50e66T + 1.24e132T^{2} \) |
| 53 | \( 1 + 4.87e67T + 1.65e136T^{2} \) |
| 59 | \( 1 + 2.11e69T + 7.89e139T^{2} \) |
| 61 | \( 1 + 5.88e70T + 1.09e141T^{2} \) |
| 67 | \( 1 + 7.00e70T + 1.81e144T^{2} \) |
| 71 | \( 1 + 8.38e72T + 1.77e146T^{2} \) |
| 73 | \( 1 + 3.60e73T + 1.59e147T^{2} \) |
| 79 | \( 1 + 1.26e75T + 8.17e149T^{2} \) |
| 83 | \( 1 + 1.04e76T + 4.04e151T^{2} \) |
| 89 | \( 1 + 8.09e76T + 1.00e154T^{2} \) |
| 97 | \( 1 + 1.98e78T + 9.01e156T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09159272985994691399228263232, −11.46194391901550885237107426326, −10.32750403528094666710094672249, −8.646664693565706599706396467910, −6.46033889647388943277533573820, −5.87407062387562113133687186334, −4.55467713696389609377004316598, −2.83226907544269284568698794055, −1.69949070065376437110594529731, 0,
1.69949070065376437110594529731, 2.83226907544269284568698794055, 4.55467713696389609377004316598, 5.87407062387562113133687186334, 6.46033889647388943277533573820, 8.646664693565706599706396467910, 10.32750403528094666710094672249, 11.46194391901550885237107426326, 13.09159272985994691399228263232
