| L(s) = 1 | − 3.26e11·2-s − 1.51e18·3-s + 6.89e22·4-s − 4.33e25·5-s + 4.94e29·6-s + 4.24e31·7-s − 1.01e34·8-s + 1.68e36·9-s + 1.41e37·10-s + 1.56e38·11-s − 1.04e41·12-s − 1.88e41·13-s − 1.38e43·14-s + 6.56e43·15-s + 7.25e44·16-s + 9.86e45·17-s − 5.50e47·18-s − 1.40e47·19-s − 2.98e48·20-s − 6.43e49·21-s − 5.11e49·22-s + 8.97e50·23-s + 1.54e52·24-s − 2.45e52·25-s + 6.15e52·26-s − 1.63e54·27-s + 2.92e54·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 1.94·3-s + 1.82·4-s − 0.266·5-s + 3.26·6-s + 0.864·7-s − 1.38·8-s + 2.77·9-s + 0.447·10-s + 0.138·11-s − 3.54·12-s − 0.317·13-s − 1.45·14-s + 0.517·15-s + 0.508·16-s + 0.711·17-s − 4.65·18-s − 0.156·19-s − 0.486·20-s − 1.67·21-s − 0.233·22-s + 0.772·23-s + 2.69·24-s − 0.929·25-s + 0.534·26-s − 3.43·27-s + 1.57·28-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
| \(L(38)\) |
\(\approx\) |
\(0.3939146891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3939146891\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.26e11T + 3.77e22T^{2} \) |
| 3 | \( 1 + 1.51e18T + 6.08e35T^{2} \) |
| 5 | \( 1 + 4.33e25T + 2.64e52T^{2} \) |
| 7 | \( 1 - 4.24e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.56e38T + 1.27e78T^{2} \) |
| 13 | \( 1 + 1.88e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 9.86e45T + 1.92e92T^{2} \) |
| 19 | \( 1 + 1.40e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 8.97e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 7.48e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 9.83e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 4.42e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.18e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 2.58e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 4.72e61T + 2.55e125T^{2} \) |
| 53 | \( 1 + 7.75e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 8.17e64T + 6.51e132T^{2} \) |
| 61 | \( 1 + 6.09e66T + 7.93e133T^{2} \) |
| 67 | \( 1 - 4.31e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 2.46e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 4.09e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 9.26e69T + 2.09e142T^{2} \) |
| 83 | \( 1 + 2.78e69T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.96e73T + 1.60e146T^{2} \) |
| 97 | \( 1 - 2.01e74T + 1.01e149T^{2} \) |
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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
−17.05176392275261344354531244194, −15.76235928230335292583310201977, −12.00951034391048467688486389958, −11.07180740004960860022466154762, −9.940437491072490005411445146049, −7.83905521870491387148149465878, −6.51922180212410194763033250794, −4.85290185031793539369583572704, −1.53357852812329393726544591288, −0.58393523381531499715284794994,
0.58393523381531499715284794994, 1.53357852812329393726544591288, 4.85290185031793539369583572704, 6.51922180212410194763033250794, 7.83905521870491387148149465878, 9.940437491072490005411445146049, 11.07180740004960860022466154762, 12.00951034391048467688486389958, 15.76235928230335292583310201977, 17.05176392275261344354531244194
