| L(s) = 1 | − 577.·2-s − 6.56e3·3-s + 2.02e5·4-s + 3.90e5·5-s + 3.79e6·6-s + 2.79e7·7-s − 4.15e7·8-s + 4.30e7·9-s − 2.25e8·10-s + 6.58e8·11-s − 1.33e9·12-s + 4.25e8·13-s − 1.61e10·14-s − 2.56e9·15-s − 2.59e9·16-s − 4.12e10·17-s − 2.48e10·18-s + 1.11e11·19-s + 7.92e10·20-s − 1.83e11·21-s − 3.80e11·22-s + 2.41e11·23-s + 2.72e11·24-s + 1.52e11·25-s − 2.45e11·26-s − 2.82e11·27-s + 5.66e12·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.54·4-s + 0.447·5-s + 0.921·6-s + 1.83·7-s − 0.875·8-s + 0.333·9-s − 0.713·10-s + 0.925·11-s − 0.893·12-s + 0.144·13-s − 2.92·14-s − 0.258·15-s − 0.151·16-s − 1.43·17-s − 0.532·18-s + 1.50·19-s + 0.692·20-s − 1.05·21-s − 1.47·22-s + 0.641·23-s + 0.505·24-s + 0.200·25-s − 0.230·26-s − 0.192·27-s + 2.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.082145275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082145275\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 + 577.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.79e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 6.58e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.25e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.12e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.11e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.41e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 9.10e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 6.61e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.53e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 9.96e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 4.20e12T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.54e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 7.54e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 6.28e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 3.95e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.34e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.00e16T + 2.96e31T^{2} \) |
| 73 | \( 1 - 8.47e14T + 4.74e31T^{2} \) |
| 79 | \( 1 - 4.80e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 6.98e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.08e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 7.39e16T + 5.95e33T^{2} \) |
| 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
−15.62813345180844813552513388172, −13.99396130219476558567858571299, −11.62466929446352938681759752987, −10.93461623467639286018115111032, −9.410470938397379270224694796583, −8.202784185795104819546644356709, −6.80052792081051672867616520788, −4.86806071729717974474124262359, −1.84046120303542545290048268367, −0.955166660030715406991007639625,
0.955166660030715406991007639625, 1.84046120303542545290048268367, 4.86806071729717974474124262359, 6.80052792081051672867616520788, 8.202784185795104819546644356709, 9.410470938397379270224694796583, 10.93461623467639286018115111032, 11.62466929446352938681759752987, 13.99396130219476558567858571299, 15.62813345180844813552513388172
