| L(s) = 1 | + 2.60e4·2-s − 1.44e7·3-s + 1.42e8·4-s − 6.21e9·5-s − 3.77e11·6-s − 3.40e10·7-s − 1.02e13·8-s + 1.41e14·9-s − 1.62e14·10-s − 7.97e14·11-s − 2.07e15·12-s + 1.10e16·13-s − 8.87e14·14-s + 9.00e16·15-s − 3.44e17·16-s − 7.47e17·17-s + 3.68e18·18-s − 2.49e18·19-s − 8.88e17·20-s + 4.93e17·21-s − 2.07e19·22-s − 1.63e18·23-s + 1.48e20·24-s − 1.47e20·25-s + 2.88e20·26-s − 1.05e21·27-s − 4.86e18·28-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 1.74·3-s + 0.266·4-s − 0.455·5-s − 1.96·6-s − 0.0189·7-s − 0.825·8-s + 2.06·9-s − 0.512·10-s − 0.633·11-s − 0.465·12-s + 0.778·13-s − 0.0213·14-s + 0.796·15-s − 1.19·16-s − 1.07·17-s + 2.31·18-s − 0.715·19-s − 0.121·20-s + 0.0331·21-s − 0.712·22-s − 0.0294·23-s + 1.44·24-s − 0.792·25-s + 0.875·26-s − 1.85·27-s − 0.00505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+29/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(15)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.60e4T + 5.36e8T^{2} \) |
| 3 | \( 1 + 1.44e7T + 6.86e13T^{2} \) |
| 5 | \( 1 + 6.21e9T + 1.86e20T^{2} \) |
| 7 | \( 1 + 3.40e10T + 3.21e24T^{2} \) |
| 11 | \( 1 + 7.97e14T + 1.58e30T^{2} \) |
| 13 | \( 1 - 1.10e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 7.47e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 2.49e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 1.63e18T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.87e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + 4.79e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 7.91e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 6.06e21T + 5.89e46T^{2} \) |
| 43 | \( 1 - 6.72e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 1.90e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.65e24T + 1.00e50T^{2} \) |
| 59 | \( 1 + 8.93e25T + 2.26e51T^{2} \) |
| 61 | \( 1 - 4.59e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 1.04e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 1.75e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + 4.56e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 3.28e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 4.40e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 6.06e26T + 3.40e56T^{2} \) |
| 97 | \( 1 - 1.44e28T + 4.13e57T^{2} \) |
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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
−23.85023141341665308092367887910, −22.99661949629539216345037865398, −21.58323327988740077110649105200, −17.98606264450376356129990123687, −15.79572792346414260372975867814, −12.86039942878683504422430358663, −11.22506451962109841988748838793, −6.12312213314869983040118889323, −4.46985241121272394578930720162, 0,
4.46985241121272394578930720162, 6.12312213314869983040118889323, 11.22506451962109841988748838793, 12.86039942878683504422430358663, 15.79572792346414260372975867814, 17.98606264450376356129990123687, 21.58323327988740077110649105200, 22.99661949629539216345037865398, 23.85023141341665308092367887910
