| L(s) = 1 | − 3.43e10·2-s + 4.72e16·3-s + 1.18e21·4-s − 7.30e24·5-s − 1.62e27·6-s − 2.33e28·7-s − 4.05e31·8-s − 5.27e33·9-s + 2.51e35·10-s − 1.13e37·11-s + 5.57e37·12-s − 4.82e39·13-s + 8.01e38·14-s − 3.45e41·15-s + 1.39e42·16-s + 5.37e43·17-s + 1.81e44·18-s + 3.28e44·19-s − 8.62e45·20-s − 1.10e45·21-s + 3.91e47·22-s + 1.06e48·23-s − 1.91e48·24-s + 1.10e49·25-s + 1.65e50·26-s − 6.04e50·27-s − 2.75e49·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.545·3-s + 0.5·4-s − 1.12·5-s − 0.385·6-s − 0.0232·7-s − 0.353·8-s − 0.702·9-s + 0.793·10-s − 1.22·11-s + 0.272·12-s − 1.37·13-s + 0.0164·14-s − 0.612·15-s + 0.250·16-s + 1.11·17-s + 0.496·18-s + 0.131·19-s − 0.561·20-s − 0.0126·21-s + 0.864·22-s + 0.484·23-s − 0.192·24-s + 0.260·25-s + 0.972·26-s − 0.928·27-s − 0.0116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(\approx\) |
\(0.6016340345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6016340345\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.43e10T \) |
| good | 3 | \( 1 - 4.72e16T + 7.50e33T^{2} \) |
| 5 | \( 1 + 7.30e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 2.33e28T + 1.00e60T^{2} \) |
| 11 | \( 1 + 1.13e37T + 8.68e73T^{2} \) |
| 13 | \( 1 + 4.82e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 5.37e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 3.28e44T + 6.18e90T^{2} \) |
| 23 | \( 1 - 1.06e48T + 4.81e96T^{2} \) |
| 29 | \( 1 + 1.14e52T + 6.76e103T^{2} \) |
| 31 | \( 1 - 1.80e52T + 7.70e105T^{2} \) |
| 37 | \( 1 + 4.58e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 2.53e57T + 3.21e114T^{2} \) |
| 43 | \( 1 - 2.43e57T + 9.46e115T^{2} \) |
| 47 | \( 1 - 5.94e58T + 5.23e118T^{2} \) |
| 53 | \( 1 + 6.61e60T + 2.65e122T^{2} \) |
| 59 | \( 1 + 1.13e63T + 5.37e125T^{2} \) |
| 61 | \( 1 - 3.97e63T + 5.73e126T^{2} \) |
| 67 | \( 1 - 1.05e65T + 4.48e129T^{2} \) |
| 71 | \( 1 - 6.61e65T + 2.75e131T^{2} \) |
| 73 | \( 1 - 1.33e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 2.53e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 2.11e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 4.02e66T + 2.55e138T^{2} \) |
| 97 | \( 1 - 3.31e70T + 1.15e141T^{2} \) |
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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
−14.52170003843116869099090047146, −12.46733284980756652870602064531, −11.14109875602125567823340233419, −9.602407719289499622842537816045, −8.076428026555780078305276314092, −7.46402528121473953105970301410, −5.29138069349547725323626923255, −3.41922911598692545131697169932, −2.35575346106003753127284087981, −0.41980751051285742381739687448,
0.41980751051285742381739687448, 2.35575346106003753127284087981, 3.41922911598692545131697169932, 5.29138069349547725323626923255, 7.46402528121473953105970301410, 8.076428026555780078305276314092, 9.602407719289499622842537816045, 11.14109875602125567823340233419, 12.46733284980756652870602064531, 14.52170003843116869099090047146
