| L(s) = 1 | + 128·2-s + 1.63e4·4-s + 3.14e5·5-s + 2.02e6·7-s + 2.09e6·8-s + 4.02e7·10-s − 1.10e8·11-s + 5.60e7·13-s + 2.59e8·14-s + 2.68e8·16-s + 1.93e9·17-s + 2.16e9·19-s + 5.15e9·20-s − 1.41e10·22-s − 6.22e9·23-s + 6.83e10·25-s + 7.17e9·26-s + 3.31e10·28-s − 6.47e10·29-s − 2.02e10·31-s + 3.43e10·32-s + 2.47e11·34-s + 6.36e11·35-s + 4.88e11·37-s + 2.76e11·38-s + 6.59e11·40-s + 7.72e11·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.80·5-s + 0.929·7-s + 0.353·8-s + 1.27·10-s − 1.70·11-s + 0.247·13-s + 0.657·14-s + 1/4·16-s + 1.14·17-s + 0.555·19-s + 0.900·20-s − 1.20·22-s − 0.381·23-s + 2.24·25-s + 0.175·26-s + 0.464·28-s − 0.696·29-s − 0.132·31-s + 0.176·32-s + 0.806·34-s + 1.67·35-s + 0.846·37-s + 0.392·38-s + 0.636·40-s + 0.619·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(4.397831836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.397831836\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{7} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 62898 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 2025056 T + p^{15} T^{2} \) |
| 11 | \( 1 + 110255052 T + p^{15} T^{2} \) |
| 13 | \( 1 - 4311374 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 1930104414 T + p^{15} T^{2} \) |
| 19 | \( 1 - 2163188180 T + p^{15} T^{2} \) |
| 23 | \( 1 + 6228974472 T + p^{15} T^{2} \) |
| 29 | \( 1 + 64743719070 T + p^{15} T^{2} \) |
| 31 | \( 1 + 20237611048 T + p^{15} T^{2} \) |
| 37 | \( 1 - 488967594446 T + p^{15} T^{2} \) |
| 41 | \( 1 - 772359114198 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1306766329292 T + p^{15} T^{2} \) |
| 47 | \( 1 + 3351821491776 T + p^{15} T^{2} \) |
| 53 | \( 1 + 9387813393702 T + p^{15} T^{2} \) |
| 59 | \( 1 + 28930359275340 T + p^{15} T^{2} \) |
| 61 | \( 1 - 42393077399702 T + p^{15} T^{2} \) |
| 67 | \( 1 + 52247243064364 T + p^{15} T^{2} \) |
| 71 | \( 1 - 27194529024648 T + p^{15} T^{2} \) |
| 73 | \( 1 + 91604195687878 T + p^{15} T^{2} \) |
| 79 | \( 1 - 62882111078120 T + p^{15} T^{2} \) |
| 83 | \( 1 - 223567315949868 T + p^{15} T^{2} \) |
| 89 | \( 1 + 554198786115210 T + p^{15} T^{2} \) |
| 97 | \( 1 + 14318252338942 p T + p^{15} T^{2} \) |
| 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
−14.66255173747013879490324824232, −13.73691039800623536557749820115, −12.77250750804935101865730088165, −10.92854797118525319698082645922, −9.790113203735723743314501488028, −7.81056503788783160259611549228, −5.85205374965197959692099114443, −5.07613467189564004284936807337, −2.71198599609019791692672971446, −1.50702914756187173951441299994,
1.50702914756187173951441299994, 2.71198599609019791692672971446, 5.07613467189564004284936807337, 5.85205374965197959692099114443, 7.81056503788783160259611549228, 9.790113203735723743314501488028, 10.92854797118525319698082645922, 12.77250750804935101865730088165, 13.73691039800623536557749820115, 14.66255173747013879490324824232
