| L(s) = 1 | + 7.00e11·2-s + 9.61e17·3-s + 3.40e23·4-s − 1.50e27·5-s + 6.73e29·6-s − 2.07e32·7-s + 1.32e35·8-s − 4.55e36·9-s − 1.05e39·10-s − 5.71e39·11-s + 3.27e41·12-s + 4.10e40·13-s − 1.45e44·14-s − 1.44e45·15-s + 4.15e46·16-s − 3.37e47·17-s − 3.18e48·18-s + 1.24e49·19-s − 5.10e50·20-s − 1.99e50·21-s − 4.00e51·22-s − 2.06e52·23-s + 1.27e53·24-s + 1.59e54·25-s + 2.87e52·26-s − 9.63e54·27-s − 7.07e55·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.410·3-s + 2.25·4-s − 1.84·5-s + 0.741·6-s − 0.604·7-s + 2.25·8-s − 0.831·9-s − 3.32·10-s − 0.460·11-s + 0.925·12-s + 0.00533·13-s − 1.08·14-s − 0.758·15-s + 1.81·16-s − 1.43·17-s − 1.49·18-s + 0.727·19-s − 4.15·20-s − 0.248·21-s − 0.831·22-s − 0.775·23-s + 0.927·24-s + 2.40·25-s + 0.00961·26-s − 0.752·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(78-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+77/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(39)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{79}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 7.00e11T + 1.51e23T^{2} \) |
| 3 | \( 1 - 9.61e17T + 5.47e36T^{2} \) |
| 5 | \( 1 + 1.50e27T + 6.61e53T^{2} \) |
| 7 | \( 1 + 2.07e32T + 1.18e65T^{2} \) |
| 11 | \( 1 + 5.71e39T + 1.53e80T^{2} \) |
| 13 | \( 1 - 4.10e40T + 5.93e85T^{2} \) |
| 17 | \( 1 + 3.37e47T + 5.55e94T^{2} \) |
| 19 | \( 1 - 1.24e49T + 2.91e98T^{2} \) |
| 23 | \( 1 + 2.06e52T + 7.12e104T^{2} \) |
| 29 | \( 1 - 8.42e55T + 4.02e112T^{2} \) |
| 31 | \( 1 - 4.46e57T + 6.83e114T^{2} \) |
| 37 | \( 1 + 7.21e59T + 5.64e120T^{2} \) |
| 41 | \( 1 + 7.96e61T + 1.52e124T^{2} \) |
| 43 | \( 1 + 2.46e62T + 5.98e125T^{2} \) |
| 47 | \( 1 + 7.65e63T + 5.64e128T^{2} \) |
| 53 | \( 1 + 1.05e66T + 5.87e132T^{2} \) |
| 59 | \( 1 + 2.11e68T + 2.26e136T^{2} \) |
| 61 | \( 1 - 1.33e68T + 2.95e137T^{2} \) |
| 67 | \( 1 - 1.91e70T + 4.05e140T^{2} \) |
| 71 | \( 1 - 7.67e70T + 3.52e142T^{2} \) |
| 73 | \( 1 - 1.11e71T + 2.99e143T^{2} \) |
| 79 | \( 1 - 1.26e73T + 1.31e146T^{2} \) |
| 83 | \( 1 - 2.46e73T + 5.87e147T^{2} \) |
| 89 | \( 1 + 5.29e74T + 1.26e150T^{2} \) |
| 97 | \( 1 + 1.16e76T + 9.58e152T^{2} \) |
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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
−15.24616970572560721974650510255, −13.71707399484211048150853382036, −12.21689780458381815623752545838, −11.22920989361848562892751142138, −8.098994977694216503554746150915, −6.59979783550000133316732570075, −4.75929443148309090120962963598, −3.59903709682804289731794552199, −2.72130235567142653526662454888, 0,
2.72130235567142653526662454888, 3.59903709682804289731794552199, 4.75929443148309090120962963598, 6.59979783550000133316732570075, 8.098994977694216503554746150915, 11.22920989361848562892751142138, 12.21689780458381815623752545838, 13.71707399484211048150853382036, 15.24616970572560721974650510255
