| L(s) = 1 | − 17.3·2-s − 815.·3-s − 7.89e3·4-s + 2.65e4·5-s + 1.41e4·6-s + 5.10e5·7-s + 2.79e5·8-s − 9.28e5·9-s − 4.60e5·10-s + 1.87e6·11-s + 6.43e6·12-s − 2.21e7·13-s − 8.86e6·14-s − 2.16e7·15-s + 5.97e7·16-s − 2.41e7·17-s + 1.61e7·18-s − 7.77e7·19-s − 2.09e8·20-s − 4.16e8·21-s − 3.26e7·22-s − 6.92e8·23-s − 2.27e8·24-s − 5.16e8·25-s + 3.85e8·26-s + 2.05e9·27-s − 4.02e9·28-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 0.646·3-s − 0.963·4-s + 0.759·5-s + 0.124·6-s + 1.63·7-s + 0.376·8-s − 0.582·9-s − 0.145·10-s + 0.319·11-s + 0.622·12-s − 1.27·13-s − 0.314·14-s − 0.490·15-s + 0.890·16-s − 0.242·17-s + 0.111·18-s − 0.379·19-s − 0.731·20-s − 1.05·21-s − 0.0613·22-s − 0.975·23-s − 0.243·24-s − 0.423·25-s + 0.244·26-s + 1.02·27-s − 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + 2.41e7T \) |
| good | 2 | \( 1 + 17.3T + 8.19e3T^{2} \) |
| 3 | \( 1 + 815.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.65e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 5.10e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.87e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.21e7T + 3.02e14T^{2} \) |
| 19 | \( 1 + 7.77e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.92e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.62e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.79e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.98e8T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.94e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.63e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 4.48e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.81e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 5.32e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 6.60e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 8.44e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 3.69e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.09e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.70e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.02e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.27e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.31e12T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.66400847115541150809363417595, −13.92880909250991883070459867075, −12.19397465634278590757670800228, −10.82887289347714162270652035060, −9.341186420192087030082061816121, −7.921260559950968357471494148945, −5.64849911918550239316481717268, −4.60932518912079572425060344188, −1.79314288870984967107414466344, 0,
1.79314288870984967107414466344, 4.60932518912079572425060344188, 5.64849911918550239316481717268, 7.921260559950968357471494148945, 9.341186420192087030082061816121, 10.82887289347714162270652035060, 12.19397465634278590757670800228, 13.92880909250991883070459867075, 14.66400847115541150809363417595
