| L(s) = 1 | + 5.85e9·2-s + 5.36e16·3-s − 2.32e21·4-s − 7.86e24·5-s + 3.14e26·6-s − 1.67e30·7-s − 2.74e31·8-s − 4.63e33·9-s − 4.60e34·10-s + 1.68e37·11-s − 1.24e38·12-s + 2.14e39·13-s − 9.80e39·14-s − 4.21e41·15-s + 5.33e42·16-s + 3.20e42·17-s − 2.71e43·18-s − 1.26e45·19-s + 1.82e46·20-s − 8.98e46·21-s + 9.84e46·22-s + 1.63e47·23-s − 1.47e48·24-s + 1.94e49·25-s + 1.25e49·26-s − 6.51e50·27-s + 3.89e51·28-s + ⋯ |
| L(s) = 1 | + 0.120·2-s + 0.619·3-s − 0.985·4-s − 1.20·5-s + 0.0745·6-s − 1.67·7-s − 0.239·8-s − 0.616·9-s − 0.145·10-s + 1.80·11-s − 0.610·12-s + 0.612·13-s − 0.201·14-s − 0.747·15-s + 0.956·16-s + 0.0668·17-s − 0.0742·18-s − 0.509·19-s + 1.19·20-s − 1.03·21-s + 0.217·22-s + 0.0745·23-s − 0.148·24-s + 0.459·25-s + 0.0737·26-s − 1.00·27-s + 1.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(72-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+71/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(36)\) |
\(\approx\) |
\(0.9866876550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9866876550\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 5.85e9T + 2.36e21T^{2} \) |
| 3 | \( 1 - 5.36e16T + 7.50e33T^{2} \) |
| 5 | \( 1 + 7.86e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 1.67e30T + 1.00e60T^{2} \) |
| 11 | \( 1 - 1.68e37T + 8.68e73T^{2} \) |
| 13 | \( 1 - 2.14e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 3.20e42T + 2.30e87T^{2} \) |
| 19 | \( 1 + 1.26e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 1.63e47T + 4.81e96T^{2} \) |
| 29 | \( 1 + 3.77e51T + 6.76e103T^{2} \) |
| 31 | \( 1 - 1.21e53T + 7.70e105T^{2} \) |
| 37 | \( 1 - 5.15e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 2.43e57T + 3.21e114T^{2} \) |
| 43 | \( 1 + 5.14e57T + 9.46e115T^{2} \) |
| 47 | \( 1 + 9.06e58T + 5.23e118T^{2} \) |
| 53 | \( 1 - 1.05e61T + 2.65e122T^{2} \) |
| 59 | \( 1 - 4.60e62T + 5.37e125T^{2} \) |
| 61 | \( 1 - 5.70e62T + 5.73e126T^{2} \) |
| 67 | \( 1 + 6.92e64T + 4.48e129T^{2} \) |
| 71 | \( 1 - 3.43e65T + 2.75e131T^{2} \) |
| 73 | \( 1 - 8.82e65T + 1.97e132T^{2} \) |
| 79 | \( 1 - 3.78e67T + 5.38e134T^{2} \) |
| 83 | \( 1 - 1.37e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 6.61e68T + 2.55e138T^{2} \) |
| 97 | \( 1 - 2.76e70T + 1.15e141T^{2} \) |
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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
−16.75188660712884310407517412250, −14.99468573613075037286802554440, −13.49595988697066186592758402709, −11.94069880127329352043172996662, −9.471664611396184743037749925597, −8.420646878646658489435212366391, −6.39152354885930146537147423028, −3.97489418947681100031575860786, −3.33263817210924801839587871020, −0.58152237278868291228749926133,
0.58152237278868291228749926133, 3.33263817210924801839587871020, 3.97489418947681100031575860786, 6.39152354885930146537147423028, 8.420646878646658489435212366391, 9.471664611396184743037749925597, 11.94069880127329352043172996662, 13.49595988697066186592758402709, 14.99468573613075037286802554440, 16.75188660712884310407517412250
