| L(s) = 1 | − 8·2-s + 64·4-s + 280.·5-s − 164.·7-s − 512·8-s − 2.24e3·10-s + 2.03e3·11-s − 1.67e3·13-s + 1.31e3·14-s + 4.09e3·16-s − 3.16e4·17-s − 1.26e4·19-s + 1.79e4·20-s − 1.63e4·22-s − 4.94e4·23-s + 513.·25-s + 1.34e4·26-s − 1.05e4·28-s + 7.55e3·29-s + 1.56e5·31-s − 3.27e4·32-s + 2.53e5·34-s − 4.60e4·35-s + 5.41e5·37-s + 1.01e5·38-s − 1.43e5·40-s − 5.37e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.00·5-s − 0.181·7-s − 0.353·8-s − 0.709·10-s + 0.461·11-s − 0.211·13-s + 0.128·14-s + 0.250·16-s − 1.56·17-s − 0.422·19-s + 0.501·20-s − 0.326·22-s − 0.847·23-s + 0.00656·25-s + 0.149·26-s − 0.0905·28-s + 0.0575·29-s + 0.940·31-s − 0.176·32-s + 1.10·34-s − 0.181·35-s + 1.75·37-s + 0.298·38-s − 0.354·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 280.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 164.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.55e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.56e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.41e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.10e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.36e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.98e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.69e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.80e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.45e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.17e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.79e6T + 8.07e13T^{2} \) |
| 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
−10.85385826934658437406856140664, −9.859959024395858474011310643770, −9.177238195058931437877851130185, −8.100473625524608215363818139229, −6.69216954773100407781264058887, −6.00620945808165988110759207462, −4.40746364399037075461460772965, −2.61286299848058691827103073186, −1.58517687163218254780867683395, 0,
1.58517687163218254780867683395, 2.61286299848058691827103073186, 4.40746364399037075461460772965, 6.00620945808165988110759207462, 6.69216954773100407781264058887, 8.100473625524608215363818139229, 9.177238195058931437877851130185, 9.859959024395858474011310643770, 10.85385826934658437406856140664
