| L(s) = 1 | − 1.00e5·2-s − 1.18e8·3-s + 1.46e9·4-s + 1.18e13·6-s − 1.07e13·7-s + 7.14e14·8-s + 8.41e15·9-s + 2.38e17·11-s − 1.73e17·12-s + 1.09e18·13-s + 1.07e18·14-s − 8.42e19·16-s − 2.82e20·17-s − 8.43e20·18-s + 4.67e20·19-s + 1.26e21·21-s − 2.38e22·22-s + 1.10e21·23-s − 8.44e22·24-s − 1.09e23·26-s − 3.37e23·27-s − 1.57e22·28-s + 1.88e24·29-s − 7.01e24·31-s + 2.31e24·32-s − 2.81e25·33-s + 2.83e25·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 1.58·3-s + 0.171·4-s + 1.71·6-s − 0.122·7-s + 0.897·8-s + 1.51·9-s + 1.56·11-s − 0.271·12-s + 0.457·13-s + 0.132·14-s − 1.14·16-s − 1.40·17-s − 1.63·18-s + 0.372·19-s + 0.193·21-s − 1.69·22-s + 0.0374·23-s − 1.42·24-s − 0.494·26-s − 0.813·27-s − 0.0208·28-s + 1.40·29-s − 1.73·31-s + 0.338·32-s − 2.47·33-s + 1.52·34-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 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
| 3 | \( 1 + 1.18e8T + 5.55e15T^{2} \) |
| 7 | \( 1 + 1.07e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.38e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.09e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 2.82e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 4.67e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 1.10e21T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.88e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 7.01e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 6.16e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 2.46e25T + 1.66e53T^{2} \) |
| 43 | \( 1 + 2.92e25T + 8.02e53T^{2} \) |
| 47 | \( 1 - 4.02e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 4.79e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.22e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 7.09e28T + 8.23e58T^{2} \) |
| 67 | \( 1 - 4.28e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 6.98e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 5.69e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 2.53e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 6.16e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.29e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 4.34e32T + 3.65e65T^{2} \) |
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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
−10.80395954223206722813032921189, −9.590432260956466861292408239092, −8.661239337931753855086444623480, −7.02454450043985181011914077205, −6.33404085834725104274438222252, −4.96071987215651894263754527598, −3.94987291465133059578203175787, −1.71009598778619755699959246847, −0.879455289649807157832991735120, 0,
0.879455289649807157832991735120, 1.71009598778619755699959246847, 3.94987291465133059578203175787, 4.96071987215651894263754527598, 6.33404085834725104274438222252, 7.02454450043985181011914077205, 8.661239337931753855086444623480, 9.590432260956466861292408239092, 10.80395954223206722813032921189
