| L(s) = 1 | + 1.15e13·2-s − 2.95e20·3-s + 9.46e25·4-s − 4.26e29·5-s − 3.41e33·6-s + 5.79e35·7-s + 6.46e38·8-s + 5.14e40·9-s − 4.92e42·10-s + 1.43e44·11-s − 2.79e46·12-s − 2.32e47·13-s + 6.69e48·14-s + 1.26e50·15-s + 3.80e51·16-s − 3.20e52·17-s + 5.94e53·18-s − 1.59e54·19-s − 4.03e55·20-s − 1.71e56·21-s + 1.65e57·22-s − 4.83e57·23-s − 1.91e59·24-s − 7.66e58·25-s − 2.68e60·26-s − 4.60e60·27-s + 5.48e61·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 1.55·3-s + 2.44·4-s − 0.838·5-s − 2.89·6-s + 0.702·7-s + 2.68·8-s + 1.43·9-s − 1.55·10-s + 0.791·11-s − 3.81·12-s − 1.05·13-s + 1.30·14-s + 1.30·15-s + 2.54·16-s − 1.62·17-s + 2.66·18-s − 0.716·19-s − 2.05·20-s − 1.09·21-s + 1.46·22-s − 0.647·23-s − 4.19·24-s − 0.296·25-s − 1.95·26-s − 0.675·27-s + 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(86-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+85/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(43)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{87}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.15e13T + 3.86e25T^{2} \) |
| 3 | \( 1 + 2.95e20T + 3.59e40T^{2} \) |
| 5 | \( 1 + 4.26e29T + 2.58e59T^{2} \) |
| 7 | \( 1 - 5.79e35T + 6.81e71T^{2} \) |
| 11 | \( 1 - 1.43e44T + 3.29e88T^{2} \) |
| 13 | \( 1 + 2.32e47T + 4.84e94T^{2} \) |
| 17 | \( 1 + 3.20e52T + 3.87e104T^{2} \) |
| 19 | \( 1 + 1.59e54T + 4.94e108T^{2} \) |
| 23 | \( 1 + 4.83e57T + 5.58e115T^{2} \) |
| 29 | \( 1 + 7.55e61T + 2.01e124T^{2} \) |
| 31 | \( 1 + 2.72e63T + 5.83e126T^{2} \) |
| 37 | \( 1 - 5.77e65T + 1.98e133T^{2} \) |
| 41 | \( 1 - 2.46e68T + 1.22e137T^{2} \) |
| 43 | \( 1 - 5.89e68T + 6.99e138T^{2} \) |
| 47 | \( 1 - 4.30e70T + 1.34e142T^{2} \) |
| 53 | \( 1 + 2.27e73T + 3.65e146T^{2} \) |
| 59 | \( 1 - 2.06e75T + 3.32e150T^{2} \) |
| 61 | \( 1 + 9.67e75T + 5.66e151T^{2} \) |
| 67 | \( 1 - 5.17e77T + 1.64e155T^{2} \) |
| 71 | \( 1 + 5.87e76T + 2.27e157T^{2} \) |
| 73 | \( 1 + 1.55e79T + 2.41e158T^{2} \) |
| 79 | \( 1 + 5.03e80T + 1.98e161T^{2} \) |
| 83 | \( 1 + 3.10e81T + 1.32e163T^{2} \) |
| 89 | \( 1 - 9.41e82T + 4.99e165T^{2} \) |
| 97 | \( 1 - 2.76e84T + 7.50e168T^{2} \) |
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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
−14.62969283987681335695165094371, −12.69841146081693057069022423293, −11.66440900762815619228932516597, −11.03465119395812355135089815801, −7.22263062580644663214449257161, −6.07670600520996162129121132472, −4.77608962739642047502159514050, −4.08999669717093132020019499283, −1.95493179288210960989945063813, 0,
1.95493179288210960989945063813, 4.08999669717093132020019499283, 4.77608962739642047502159514050, 6.07670600520996162129121132472, 7.22263062580644663214449257161, 11.03465119395812355135089815801, 11.66440900762815619228932516597, 12.69841146081693057069022423293, 14.62969283987681335695165094371
