| L(s) = 1 | − 8·2-s − 4.34·3-s + 64·4-s + 479.·5-s + 34.7·6-s − 1.13e3·7-s − 512·8-s − 2.16e3·9-s − 3.83e3·10-s + 7.66e3·11-s − 277.·12-s + 5.04e3·13-s + 9.11e3·14-s − 2.08e3·15-s + 4.09e3·16-s + 1.71e4·17-s + 1.73e4·18-s − 6.85e3·19-s + 3.07e4·20-s + 4.94e3·21-s − 6.13e4·22-s + 7.11e4·23-s + 2.22e3·24-s + 1.52e5·25-s − 4.03e4·26-s + 1.89e4·27-s − 7.28e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0928·3-s + 0.5·4-s + 1.71·5-s + 0.0656·6-s − 1.25·7-s − 0.353·8-s − 0.991·9-s − 1.21·10-s + 1.73·11-s − 0.0464·12-s + 0.637·13-s + 0.887·14-s − 0.159·15-s + 0.250·16-s + 0.847·17-s + 0.701·18-s − 0.229·19-s + 0.858·20-s + 0.116·21-s − 1.22·22-s + 1.21·23-s + 0.0328·24-s + 1.94·25-s − 0.450·26-s + 0.184·27-s − 0.627·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.549796479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549796479\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 + 4.34T + 2.18e3T^{2} \) |
| 5 | \( 1 - 479.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.13e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.04e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.71e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 7.11e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.22e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.44e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 120.T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.11e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.87e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.58e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.30e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.02e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.12e7T + 8.07e13T^{2} \) |
| 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
−14.64207489531025652892322361865, −13.67897111515672893487798562660, −12.34989896592906312005896818868, −10.79301479765061739442322805669, −9.506723074617059862670597511685, −8.979934108385818725973002491780, −6.57715966607988348260165599917, −5.91709819697111846823935045847, −2.99613492014459687651056813198, −1.17816343943625185396063877845,
1.17816343943625185396063877845, 2.99613492014459687651056813198, 5.91709819697111846823935045847, 6.57715966607988348260165599917, 8.979934108385818725973002491780, 9.506723074617059862670597511685, 10.79301479765061739442322805669, 12.34989896592906312005896818868, 13.67897111515672893487798562660, 14.64207489531025652892322361865
