| L(s) = 1 | − 3.27e4·2-s − 2.68e7·3-s + 1.07e9·4-s + 3.05e10·5-s + 8.78e11·6-s + 7.28e11·7-s − 3.51e13·8-s + 1.01e14·9-s − 1.00e15·10-s − 3.37e15·11-s − 2.87e16·12-s − 1.37e17·13-s − 2.38e16·14-s − 8.18e17·15-s + 1.15e18·16-s + 2.39e18·17-s − 3.31e18·18-s + 9.48e18·19-s + 3.27e19·20-s − 1.95e19·21-s + 1.10e20·22-s + 1.89e21·23-s + 9.43e20·24-s + 9.31e20·25-s + 4.50e21·26-s + 1.38e22·27-s + 7.82e20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.07·3-s + 0.5·4-s + 0.447·5-s + 0.762·6-s + 0.0580·7-s − 0.353·8-s + 0.164·9-s − 0.316·10-s − 0.243·11-s − 0.539·12-s − 0.745·13-s − 0.0410·14-s − 0.482·15-s + 0.250·16-s + 0.203·17-s − 0.115·18-s + 0.143·19-s + 0.223·20-s − 0.0625·21-s + 0.172·22-s + 1.48·23-s + 0.381·24-s + 0.200·25-s + 0.527·26-s + 0.901·27-s + 0.0290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| 5 | \( 1 - 3.05e10T \) |
| good | 3 | \( 1 + 2.68e7T + 6.17e14T^{2} \) |
| 7 | \( 1 - 7.28e11T + 1.57e26T^{2} \) |
| 11 | \( 1 + 3.37e15T + 1.91e32T^{2} \) |
| 13 | \( 1 + 1.37e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 2.39e18T + 1.39e38T^{2} \) |
| 19 | \( 1 - 9.48e18T + 4.37e39T^{2} \) |
| 23 | \( 1 - 1.89e21T + 1.63e42T^{2} \) |
| 29 | \( 1 - 5.03e21T + 2.15e45T^{2} \) |
| 31 | \( 1 + 8.54e22T + 1.70e46T^{2} \) |
| 37 | \( 1 - 1.26e24T + 4.11e48T^{2} \) |
| 41 | \( 1 - 8.53e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 2.40e25T + 4.34e50T^{2} \) |
| 47 | \( 1 + 3.96e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 7.23e25T + 2.83e53T^{2} \) |
| 59 | \( 1 + 1.71e27T + 7.87e54T^{2} \) |
| 61 | \( 1 - 2.02e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 2.22e28T + 4.05e56T^{2} \) |
| 71 | \( 1 + 8.76e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 6.23e28T + 5.79e57T^{2} \) |
| 79 | \( 1 - 1.07e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 5.93e28T + 3.10e59T^{2} \) |
| 89 | \( 1 - 1.10e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 5.31e30T + 3.88e61T^{2} \) |
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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
−12.77433843909785573955697051680, −11.47537877094027112145931539672, −10.45006599346610319078834593893, −9.131936585573792538904487484507, −7.41210979656458481029895846295, −6.08891750021286976143166536811, −4.95846014527165799493772555322, −2.73285973627266801187767919575, −1.15784275083442248784408547920, 0,
1.15784275083442248784408547920, 2.73285973627266801187767919575, 4.95846014527165799493772555322, 6.08891750021286976143166536811, 7.41210979656458481029895846295, 9.131936585573792538904487484507, 10.45006599346610319078834593893, 11.47537877094027112145931539672, 12.77433843909785573955697051680
