| L(s) = 1 | + 4.96e4·2-s − 1.55e7·3-s − 6.12e9·4-s + 2.04e10·5-s − 7.71e11·6-s − 1.22e14·7-s − 7.30e14·8-s − 5.31e15·9-s + 1.01e15·10-s + 2.15e17·11-s + 9.50e16·12-s − 1.07e18·13-s − 6.06e18·14-s − 3.17e17·15-s + 1.62e19·16-s + 2.54e20·17-s − 2.64e20·18-s − 1.22e21·19-s − 1.25e20·20-s + 1.89e21·21-s + 1.07e22·22-s − 5.11e21·23-s + 1.13e22·24-s − 1.15e23·25-s − 5.35e22·26-s + 1.68e23·27-s + 7.47e23·28-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 0.208·3-s − 0.712·4-s + 0.0598·5-s − 0.111·6-s − 1.38·7-s − 0.918·8-s − 0.956·9-s + 0.0320·10-s + 1.41·11-s + 0.148·12-s − 0.449·13-s − 0.744·14-s − 0.0124·15-s + 0.220·16-s + 1.26·17-s − 0.512·18-s − 0.970·19-s − 0.0426·20-s + 0.289·21-s + 0.758·22-s − 0.173·23-s + 0.191·24-s − 0.996·25-s − 0.240·26-s + 0.407·27-s + 0.990·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(34-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+33/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 4.96e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 1.55e7T + 5.55e15T^{2} \) |
| 5 | \( 1 - 2.04e10T + 1.16e23T^{2} \) |
| 7 | \( 1 + 1.22e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.15e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.07e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.54e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.22e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 5.11e21T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.64e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 6.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 7.46e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 4.96e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.99e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 2.16e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 3.60e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.87e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.18e27T + 8.23e58T^{2} \) |
| 67 | \( 1 - 4.85e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.42e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 7.01e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.95e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.23e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 7.05e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.71e32T + 3.65e65T^{2} \) |
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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
−23.17164710223873043569343353110, −22.08212852190947044930996024239, −19.37737814171362848300641710869, −16.99729322981038600458338655937, −14.33154534246216629846064850558, −12.38824383391611598453844590283, −9.337420969687700231085444375144, −5.97786009657894493138722886448, −3.56400685368044213449625106689, 0,
3.56400685368044213449625106689, 5.97786009657894493138722886448, 9.337420969687700231085444375144, 12.38824383391611598453844590283, 14.33154534246216629846064850558, 16.99729322981038600458338655937, 19.37737814171362848300641710869, 22.08212852190947044930996024239, 23.17164710223873043569343353110
