| L(s) = 1 | − 9·3-s + 20·5-s + 122·7-s + 81·9-s + 724·11-s + 914·13-s − 180·15-s + 1.00e3·17-s − 2.92e3·19-s − 1.09e3·21-s + 3.12e3·23-s − 2.72e3·25-s − 729·27-s − 6.74e3·29-s + 5.01e3·31-s − 6.51e3·33-s + 2.44e3·35-s + 5.27e3·37-s − 8.22e3·39-s + 5.23e3·41-s + 1.67e4·43-s + 1.62e3·45-s + 1.10e3·47-s − 1.92e3·49-s − 9.05e3·51-s − 2.20e4·53-s + 1.44e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.357·5-s + 0.941·7-s + 1/3·9-s + 1.80·11-s + 1.49·13-s − 0.206·15-s + 0.844·17-s − 1.85·19-s − 0.543·21-s + 1.23·23-s − 0.871·25-s − 0.192·27-s − 1.48·29-s + 0.936·31-s − 1.04·33-s + 0.336·35-s + 0.633·37-s − 0.866·39-s + 0.486·41-s + 1.38·43-s + 0.119·45-s + 0.0731·47-s − 0.114·49-s − 0.487·51-s − 1.07·53-s + 0.645·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.648858549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648858549\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 122 T + p^{5} T^{2} \) |
| 11 | \( 1 - 724 T + p^{5} T^{2} \) |
| 13 | \( 1 - 914 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1006 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2920 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3124 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6744 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5010 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5278 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5238 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16752 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1108 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22008 T + p^{5} T^{2} \) |
| 59 | \( 1 + 23716 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45202 T + p^{5} T^{2} \) |
| 67 | \( 1 - 22756 T + p^{5} T^{2} \) |
| 71 | \( 1 - 53436 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4790 T + p^{5} T^{2} \) |
| 79 | \( 1 + 1886 T + p^{5} T^{2} \) |
| 83 | \( 1 - 11268 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73522 T + p^{5} T^{2} \) |
| 97 | \( 1 - 114154 T + p^{5} T^{2} \) |
| 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
−10.94174973424841011588119552391, −9.515352116178133413719414016373, −8.778180345968386904228776706031, −7.73619363980800688328112366032, −6.41005774754617436420764528857, −5.94636828558007590754767705994, −4.56351164952513179229819917889, −3.71862259737100478616604103635, −1.78888328152963311681069028790, −0.998898799936081626007426931831,
0.998898799936081626007426931831, 1.78888328152963311681069028790, 3.71862259737100478616604103635, 4.56351164952513179229819917889, 5.94636828558007590754767705994, 6.41005774754617436420764528857, 7.73619363980800688328112366032, 8.778180345968386904228776706031, 9.515352116178133413719414016373, 10.94174973424841011588119552391
