| L(s) = 1 | + 8·2-s + 64·4-s − 385·5-s − 293·7-s + 512·8-s − 3.08e3·10-s + 5.40e3·11-s + 2.19e3·13-s − 2.34e3·14-s + 4.09e3·16-s + 2.10e4·17-s − 2.73e4·19-s − 2.46e4·20-s + 4.32e4·22-s + 6.30e4·23-s + 7.01e4·25-s + 1.75e4·26-s − 1.87e4·28-s − 1.22e5·29-s − 2.08e5·31-s + 3.27e4·32-s + 1.68e5·34-s + 1.12e5·35-s − 4.42e5·37-s − 2.18e5·38-s − 1.97e5·40-s − 5.80e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.37·5-s − 0.322·7-s + 0.353·8-s − 0.973·10-s + 1.22·11-s + 0.277·13-s − 0.228·14-s + 1/4·16-s + 1.03·17-s − 0.913·19-s − 0.688·20-s + 0.865·22-s + 1.08·23-s + 0.897·25-s + 0.196·26-s − 0.161·28-s − 0.930·29-s − 1.25·31-s + 0.176·32-s + 0.733·34-s + 0.444·35-s − 1.43·37-s − 0.646·38-s − 0.486·40-s − 0.131·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 77 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 293 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5402 T + p^{7} T^{2} \) |
| 17 | \( 1 - 21011 T + p^{7} T^{2} \) |
| 19 | \( 1 + 27326 T + p^{7} T^{2} \) |
| 23 | \( 1 - 63072 T + p^{7} T^{2} \) |
| 29 | \( 1 + 122238 T + p^{7} T^{2} \) |
| 31 | \( 1 + 208396 T + p^{7} T^{2} \) |
| 37 | \( 1 + 442379 T + p^{7} T^{2} \) |
| 41 | \( 1 + 58000 T + p^{7} T^{2} \) |
| 43 | \( 1 + 202025 T + p^{7} T^{2} \) |
| 47 | \( 1 + 588511 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1684336 T + p^{7} T^{2} \) |
| 59 | \( 1 - 442630 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1083608 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3443486 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2084705 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5937890 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6609256 T + p^{7} T^{2} \) |
| 83 | \( 1 - 142740 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6985286 T + p^{7} T^{2} \) |
| 97 | \( 1 + 200762 T + p^{7} T^{2} \) |
| 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
−10.83773553186241876316342095511, −9.412089970935848789872661434446, −8.318286734081370225938661702715, −7.27608039087104508238097108088, −6.43730867606195281079688031895, −5.07639553935898700318013842853, −3.84739987799775624175680934771, −3.36413228845551976686974094589, −1.49868688253634829475636753742, 0,
1.49868688253634829475636753742, 3.36413228845551976686974094589, 3.84739987799775624175680934771, 5.07639553935898700318013842853, 6.43730867606195281079688031895, 7.27608039087104508238097108088, 8.318286734081370225938661702715, 9.412089970935848789872661434446, 10.83773553186241876316342095511
