| L(s) = 1 | − 8·2-s − 82·3-s + 64·4-s + 448·5-s + 656·6-s − 343·7-s − 512·8-s + 4.53e3·9-s − 3.58e3·10-s + 2.40e3·11-s − 5.24e3·12-s + 7.11e3·13-s + 2.74e3·14-s − 3.67e4·15-s + 4.09e3·16-s + 2.48e3·17-s − 3.62e4·18-s + 3.64e4·19-s + 2.86e4·20-s + 2.81e4·21-s − 1.92e4·22-s − 1.28e4·23-s + 4.19e4·24-s + 1.22e5·25-s − 5.69e4·26-s − 1.92e5·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.75·3-s + 1/2·4-s + 1.60·5-s + 1.23·6-s − 0.377·7-s − 0.353·8-s + 2.07·9-s − 1.13·10-s + 0.545·11-s − 0.876·12-s + 0.898·13-s + 0.267·14-s − 2.81·15-s + 1/4·16-s + 0.122·17-s − 1.46·18-s + 1.22·19-s + 0.801·20-s + 0.662·21-s − 0.385·22-s − 0.220·23-s + 0.619·24-s + 1.56·25-s − 0.635·26-s − 1.88·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8307900383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8307900383\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 82 T + p^{7} T^{2} \) |
| 5 | \( 1 - 448 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2408 T + p^{7} T^{2} \) |
| 13 | \( 1 - 7116 T + p^{7} T^{2} \) |
| 17 | \( 1 - 2486 T + p^{7} T^{2} \) |
| 19 | \( 1 - 36482 T + p^{7} T^{2} \) |
| 23 | \( 1 + 560 p T + p^{7} T^{2} \) |
| 29 | \( 1 + 88094 T + p^{7} T^{2} \) |
| 31 | \( 1 - 282636 T + p^{7} T^{2} \) |
| 37 | \( 1 + 214534 T + p^{7} T^{2} \) |
| 41 | \( 1 + 140874 T + p^{7} T^{2} \) |
| 43 | \( 1 - 848 p T + p^{7} T^{2} \) |
| 47 | \( 1 - 716868 T + p^{7} T^{2} \) |
| 53 | \( 1 + 56946 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2149862 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3084360 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3034364 T + p^{7} T^{2} \) |
| 71 | \( 1 + 106624 T + p^{7} T^{2} \) |
| 73 | \( 1 - 988930 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3415896 T + p^{7} T^{2} \) |
| 83 | \( 1 + 15142 T + p^{7} T^{2} \) |
| 89 | \( 1 - 174810 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13506790 T + p^{7} T^{2} \) |
| 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
−17.73610505753424173474109583908, −17.02950834184976504205130192613, −15.96849769400455349787404246440, −13.56339706571102621655797947069, −11.97413054261582370033651680060, −10.57689597773280735553669507498, −9.531843796425403344433679608492, −6.59539017034699864580637435114, −5.62616050802270837992013920583, −1.19385621467107446928197029499,
1.19385621467107446928197029499, 5.62616050802270837992013920583, 6.59539017034699864580637435114, 9.531843796425403344433679608492, 10.57689597773280735553669507498, 11.97413054261582370033651680060, 13.56339706571102621655797947069, 15.96849769400455349787404246440, 17.02950834184976504205130192613, 17.73610505753424173474109583908
