| L(s) = 1 | − 1.37e11·2-s + 1.40e18·3-s + 1.88e22·4-s + 2.68e26·5-s − 1.92e29·6-s + 2.35e31·7-s − 2.59e33·8-s + 1.35e36·9-s − 3.69e37·10-s − 4.78e38·11-s + 2.65e40·12-s − 9.23e41·13-s − 3.23e42·14-s + 3.76e44·15-s + 3.56e44·16-s + 2.36e46·17-s − 1.86e47·18-s − 1.55e47·19-s + 5.07e48·20-s + 3.30e49·21-s + 6.58e49·22-s + 1.11e51·23-s − 3.64e51·24-s + 4.57e52·25-s + 1.26e53·26-s + 1.05e54·27-s + 4.44e53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.79·3-s + 0.5·4-s + 1.65·5-s − 1.27·6-s + 0.479·7-s − 0.353·8-s + 2.23·9-s − 1.16·10-s − 0.424·11-s + 0.899·12-s − 1.55·13-s − 0.338·14-s + 2.97·15-s + 0.250·16-s + 1.70·17-s − 1.58·18-s − 0.173·19-s + 0.825·20-s + 0.861·21-s + 0.300·22-s + 0.956·23-s − 0.635·24-s + 1.72·25-s + 1.10·26-s + 2.22·27-s + 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(4.605626545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.605626545\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.37e11T \) |
| good | 3 | \( 1 - 1.40e18T + 6.08e35T^{2} \) |
| 5 | \( 1 - 2.68e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 2.35e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 4.78e38T + 1.27e78T^{2} \) |
| 13 | \( 1 + 9.23e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 2.36e46T + 1.92e92T^{2} \) |
| 19 | \( 1 + 1.55e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 1.11e51T + 1.34e102T^{2} \) |
| 29 | \( 1 - 2.09e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 1.05e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 1.17e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 1.22e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 1.51e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 4.28e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 4.22e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 2.77e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 4.36e66T + 7.93e133T^{2} \) |
| 67 | \( 1 - 3.20e67T + 9.02e136T^{2} \) |
| 71 | \( 1 - 4.96e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 2.99e68T + 5.61e139T^{2} \) |
| 79 | \( 1 + 7.25e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 1.00e72T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.72e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 5.00e74T + 1.01e149T^{2} \) |
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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
−14.14175726747458754280584060512, −12.78219570230413558532329475023, −10.08535854157131868080875053716, −9.567747672167328515128594827500, −8.288330783217853007326929080582, −7.14142206508146716606849447384, −5.17664645352531999028967262865, −2.99491477356217662861848751487, −2.20741526402792460655730107632, −1.28989797488801820929982837747,
1.28989797488801820929982837747, 2.20741526402792460655730107632, 2.99491477356217662861848751487, 5.17664645352531999028967262865, 7.14142206508146716606849447384, 8.288330783217853007326929080582, 9.567747672167328515128594827500, 10.08535854157131868080875053716, 12.78219570230413558532329475023, 14.14175726747458754280584060512
