| L(s) = 1 | − 5.62e3·2-s + 1.24e8·3-s − 8.55e9·4-s − 7.01e11·6-s − 7.01e13·7-s + 9.65e13·8-s + 9.96e15·9-s + 9.39e16·11-s − 1.06e18·12-s − 9.95e17·13-s + 3.94e17·14-s + 7.29e19·16-s + 2.59e20·17-s − 5.60e19·18-s − 1.31e21·19-s − 8.74e21·21-s − 5.28e20·22-s − 5.44e22·23-s + 1.20e22·24-s + 5.60e21·26-s + 5.49e23·27-s + 6.00e23·28-s − 9.77e23·29-s + 6.82e24·31-s − 1.23e24·32-s + 1.17e25·33-s − 1.45e24·34-s + ⋯ |
| L(s) = 1 | − 0.0607·2-s + 1.67·3-s − 0.996·4-s − 0.101·6-s − 0.797·7-s + 0.121·8-s + 1.79·9-s + 0.616·11-s − 1.66·12-s − 0.414·13-s + 0.0484·14-s + 0.988·16-s + 1.29·17-s − 0.108·18-s − 1.04·19-s − 1.33·21-s − 0.0374·22-s − 1.85·23-s + 0.202·24-s + 0.0251·26-s + 1.32·27-s + 0.794·28-s − 0.725·29-s + 1.68·31-s − 0.181·32-s + 1.03·33-s − 0.0784·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 5.62e3T + 8.58e9T^{2} \) |
| 3 | \( 1 - 1.24e8T + 5.55e15T^{2} \) |
| 7 | \( 1 + 7.01e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 9.39e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 9.95e17T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.59e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.31e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 5.44e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 9.77e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 6.82e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 7.50e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 2.94e25T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.00e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 1.91e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 4.33e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.21e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.03e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 1.33e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 4.10e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 8.64e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.76e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 6.43e30T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.32e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 2.90e32T + 3.65e65T^{2} \) |
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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
−9.959898167075795547439679919649, −9.576749758250922501893848687393, −8.440159216433238149050444788815, −7.71522971227882952024329997564, −6.12599235002385305446207432381, −4.35641843699847678518091001050, −3.63768308086021634292186016006, −2.63751404917680081658840823328, −1.36505051645177571274278094766, 0,
1.36505051645177571274278094766, 2.63751404917680081658840823328, 3.63768308086021634292186016006, 4.35641843699847678518091001050, 6.12599235002385305446207432381, 7.71522971227882952024329997564, 8.440159216433238149050444788815, 9.576749758250922501893848687393, 9.959898167075795547439679919649
