| L(s) = 1 | + 1.37e11·2-s + 7.85e17·3-s + 1.88e22·4-s + 1.02e26·5-s + 1.07e29·6-s − 9.01e31·7-s + 2.59e33·8-s + 8.79e33·9-s + 1.40e37·10-s + 1.29e39·11-s + 1.48e40·12-s − 8.54e41·13-s − 1.23e43·14-s + 8.05e43·15-s + 3.56e44·16-s − 1.96e46·17-s + 1.20e45·18-s + 1.45e47·19-s + 1.93e48·20-s − 7.08e49·21-s + 1.78e50·22-s − 9.62e50·23-s + 2.03e51·24-s − 1.59e52·25-s − 1.17e53·26-s − 4.70e53·27-s − 1.70e54·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s + 0.630·5-s + 0.712·6-s − 1.83·7-s + 0.353·8-s + 0.0144·9-s + 0.445·10-s + 1.15·11-s + 0.503·12-s − 1.44·13-s − 1.29·14-s + 0.634·15-s + 0.250·16-s − 1.42·17-s + 0.0102·18-s + 0.161·19-s + 0.315·20-s − 1.84·21-s + 0.813·22-s − 0.829·23-s + 0.356·24-s − 0.602·25-s − 1.01·26-s − 0.992·27-s − 0.918·28-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
| \(L(38)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{77}{2})\) |
|
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 | \( 1 - 7.85e17T + 6.08e35T^{2} \) |
| 5 | \( 1 - 1.02e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 9.01e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.29e39T + 1.27e78T^{2} \) |
| 13 | \( 1 + 8.54e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.96e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.45e47T + 8.06e95T^{2} \) |
| 23 | \( 1 + 9.62e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 6.83e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 2.27e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 6.31e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.47e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 1.32e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 4.14e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 1.50e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 1.88e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 2.68e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 3.70e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 2.25e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 2.96e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 2.41e71T + 2.09e142T^{2} \) |
| 83 | \( 1 - 9.02e70T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.80e73T + 1.60e146T^{2} \) |
| 97 | \( 1 - 5.17e74T + 1.01e149T^{2} \) |
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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
−13.39143704668627239759294778899, −12.14118921017957122054885485528, −9.925440628563798589744946197335, −9.049209669254344947630128362586, −7.02288618389744931570776488379, −6.00740095982595315833604324869, −4.10340663292461747201019682663, −2.94908652091145676133944266262, −2.10387325063537903917906818028, 0,
2.10387325063537903917906818028, 2.94908652091145676133944266262, 4.10340663292461747201019682663, 6.00740095982595315833604324869, 7.02288618389744931570776488379, 9.049209669254344947630128362586, 9.925440628563798589744946197335, 12.14118921017957122054885485528, 13.39143704668627239759294778899
