| L(s) = 1 | + 49.4·5-s + 93.1·7-s − 173.·11-s + 219.·17-s + 361·19-s + 158·23-s + 1.82e3·25-s + 4.60e3·35-s − 800.·43-s − 3.07e3·47-s + 6.27e3·49-s − 8.56e3·55-s − 7.41e3·61-s + 1.90e3·73-s − 1.61e4·77-s + 5.67e3·83-s + 1.08e4·85-s + 1.78e4·95-s + 9.99e3·101-s + 7.81e3·115-s + 2.04e4·119-s + ⋯ |
| L(s) = 1 | + 1.97·5-s + 1.90·7-s − 1.43·11-s + 0.760·17-s + 19-s + 0.298·23-s + 2.91·25-s + 3.76·35-s − 0.433·43-s − 1.39·47-s + 2.61·49-s − 2.83·55-s − 1.99·61-s + 0.356·73-s − 2.71·77-s + 0.824·83-s + 1.50·85-s + 1.97·95-s + 0.980·101-s + 0.591·115-s + 1.44·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.136445671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.136445671\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 5 | \( 1 - 49.4T + 625T^{2} \) |
| 7 | \( 1 - 93.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 173.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 219.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 158T + 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 800.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.07e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.41e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.90e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 5.67e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
| 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
−10.02253749190451278162832118078, −9.097713414974647512888263573643, −8.140664795171600896140142054606, −7.41380428436601002106888725730, −6.07374671003484679137551808463, −5.14410936544971524311460648739, −4.99733779350976551708115986404, −2.91715570505957179991071444979, −1.94529827620458434947262888100, −1.18351352672734375070448118228,
1.18351352672734375070448118228, 1.94529827620458434947262888100, 2.91715570505957179991071444979, 4.99733779350976551708115986404, 5.14410936544971524311460648739, 6.07374671003484679137551808463, 7.41380428436601002106888725730, 8.140664795171600896140142054606, 9.097713414974647512888263573643, 10.02253749190451278162832118078
