| L(s) = 1 | + 1.28e10·2-s − 4.48e17·3-s − 9.28e21·4-s + 4.05e25·5-s − 5.74e27·6-s − 7.12e30·7-s − 2.39e32·8-s + 1.33e35·9-s + 5.19e35·10-s + 5.42e37·11-s + 4.16e39·12-s + 7.16e40·13-s − 9.12e40·14-s − 1.82e43·15-s + 8.45e43·16-s + 1.30e44·17-s + 1.71e45·18-s − 2.65e46·19-s − 3.76e47·20-s + 3.19e48·21-s + 6.94e47·22-s − 6.34e49·23-s + 1.07e50·24-s + 5.89e50·25-s + 9.17e50·26-s − 2.97e52·27-s + 6.61e52·28-s + ⋯ |
| L(s) = 1 | + 0.131·2-s − 1.72·3-s − 0.982·4-s + 1.24·5-s − 0.227·6-s − 1.01·7-s − 0.261·8-s + 1.98·9-s + 0.164·10-s + 0.529·11-s + 1.69·12-s + 1.57·13-s − 0.133·14-s − 2.15·15-s + 0.948·16-s + 0.160·17-s + 0.260·18-s − 0.561·19-s − 1.22·20-s + 1.75·21-s + 0.0697·22-s − 1.25·23-s + 0.451·24-s + 0.556·25-s + 0.206·26-s − 1.69·27-s + 0.997·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(37)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.28e10T + 9.44e21T^{2} \) |
| 3 | \( 1 + 4.48e17T + 6.75e34T^{2} \) |
| 5 | \( 1 - 4.05e25T + 1.05e51T^{2} \) |
| 7 | \( 1 + 7.12e30T + 4.92e61T^{2} \) |
| 11 | \( 1 - 5.42e37T + 1.05e76T^{2} \) |
| 13 | \( 1 - 7.16e40T + 2.07e81T^{2} \) |
| 17 | \( 1 - 1.30e44T + 6.64e89T^{2} \) |
| 19 | \( 1 + 2.65e46T + 2.23e93T^{2} \) |
| 23 | \( 1 + 6.34e49T + 2.54e99T^{2} \) |
| 29 | \( 1 - 8.00e52T + 5.68e106T^{2} \) |
| 31 | \( 1 + 3.53e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 1.61e56T + 3.01e114T^{2} \) |
| 41 | \( 1 - 6.62e58T + 5.41e117T^{2} \) |
| 43 | \( 1 - 5.13e57T + 1.75e119T^{2} \) |
| 47 | \( 1 - 1.87e61T + 1.15e122T^{2} \) |
| 53 | \( 1 + 2.36e62T + 7.44e125T^{2} \) |
| 59 | \( 1 + 2.19e64T + 1.87e129T^{2} \) |
| 61 | \( 1 + 1.30e65T + 2.13e130T^{2} \) |
| 67 | \( 1 + 5.38e66T + 2.01e133T^{2} \) |
| 71 | \( 1 - 3.88e67T + 1.38e135T^{2} \) |
| 73 | \( 1 + 9.28e67T + 1.05e136T^{2} \) |
| 79 | \( 1 + 4.44e68T + 3.36e138T^{2} \) |
| 83 | \( 1 - 9.20e69T + 1.23e140T^{2} \) |
| 89 | \( 1 + 3.27e70T + 2.02e142T^{2} \) |
| 97 | \( 1 + 2.09e72T + 1.08e145T^{2} \) |
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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
−16.32397618486113917069065523289, −13.63431323517529663649049601018, −12.50701869284297717295949755892, −10.55945348728993121237476851858, −9.336242290136873608398234029104, −6.24241685386205450248069218885, −5.73303297228531362946256654245, −4.01676214092327570276345851221, −1.28192350893123960744080206343, 0,
1.28192350893123960744080206343, 4.01676214092327570276345851221, 5.73303297228531362946256654245, 6.24241685386205450248069218885, 9.336242290136873608398234029104, 10.55945348728993121237476851858, 12.50701869284297717295949755892, 13.63431323517529663649049601018, 16.32397618486113917069065523289
