| L(s) = 1 | − 36·3-s − 125·5-s − 776·7-s − 891·9-s − 124·11-s + 1.30e4·13-s + 4.50e3·15-s − 1.59e4·17-s − 2.05e4·19-s + 2.79e4·21-s + 2.92e4·23-s + 1.56e4·25-s + 1.10e5·27-s + 2.21e5·29-s + 1.09e5·31-s + 4.46e3·33-s + 9.70e4·35-s − 7.34e4·37-s − 4.70e5·39-s + 1.27e4·41-s + 2.90e5·43-s + 1.11e5·45-s − 1.26e6·47-s − 2.21e5·49-s + 5.74e5·51-s + 3.95e5·53-s + 1.55e4·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s − 0.855·7-s − 0.407·9-s − 0.0280·11-s + 1.65·13-s + 0.344·15-s − 0.787·17-s − 0.686·19-s + 0.658·21-s + 0.500·23-s + 1/5·25-s + 1.08·27-s + 1.68·29-s + 0.661·31-s + 0.0216·33-s + 0.382·35-s − 0.238·37-s − 1.27·39-s + 0.0289·41-s + 0.557·43-s + 0.182·45-s − 1.78·47-s − 0.268·49-s + 0.606·51-s + 0.365·53-s + 0.0125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 4 p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 + 776 T + p^{7} T^{2} \) |
| 11 | \( 1 + 124 T + p^{7} T^{2} \) |
| 13 | \( 1 - 13082 T + p^{7} T^{2} \) |
| 17 | \( 1 + 15950 T + p^{7} T^{2} \) |
| 19 | \( 1 + 20516 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29224 T + p^{7} T^{2} \) |
| 29 | \( 1 - 221482 T + p^{7} T^{2} \) |
| 31 | \( 1 - 109760 T + p^{7} T^{2} \) |
| 37 | \( 1 + 73422 T + p^{7} T^{2} \) |
| 41 | \( 1 - 12762 T + p^{7} T^{2} \) |
| 43 | \( 1 - 290548 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1269152 T + p^{7} T^{2} \) |
| 53 | \( 1 - 395778 T + p^{7} T^{2} \) |
| 59 | \( 1 - 421492 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2122250 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3132868 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5376552 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4985466 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3867504 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6190196 T + p^{7} T^{2} \) |
| 89 | \( 1 - 1124394 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9968098 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18451172193420730301282962439, −8.864449931761722349374310298544, −8.248646476488909788129710408552, −6.56489573012369598003575965255, −6.35293802998588689586245964710, −5.02587240672655900031024315507, −3.86085294622091494107535869023, −2.77888168058026215233205854725, −1.01897062916886156883320648197, 0,
1.01897062916886156883320648197, 2.77888168058026215233205854725, 3.86085294622091494107535869023, 5.02587240672655900031024315507, 6.35293802998588689586245964710, 6.56489573012369598003575965255, 8.248646476488909788129710408552, 8.864449931761722349374310298544, 10.18451172193420730301282962439
