| L(s) = 1 | − 8·2-s − 19·3-s + 64·4-s + 317·5-s + 152·6-s − 1.03e3·7-s − 512·8-s − 1.82e3·9-s − 2.53e3·10-s + 1.33e3·11-s − 1.21e3·12-s − 1.46e4·13-s + 8.24e3·14-s − 6.02e3·15-s + 4.09e3·16-s − 3.00e4·17-s + 1.46e4·18-s + 3.80e4·19-s + 2.02e4·20-s + 1.95e4·21-s − 1.06e4·22-s − 1.29e4·23-s + 9.72e3·24-s + 2.23e4·25-s + 1.17e5·26-s + 7.62e4·27-s − 6.59e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.406·3-s + 1/2·4-s + 1.13·5-s + 0.287·6-s − 1.13·7-s − 0.353·8-s − 0.834·9-s − 0.801·10-s + 0.301·11-s − 0.203·12-s − 1.85·13-s + 0.802·14-s − 0.460·15-s + 1/4·16-s − 1.48·17-s + 0.590·18-s + 1.27·19-s + 0.567·20-s + 0.461·21-s − 0.213·22-s − 0.221·23-s + 0.143·24-s + 0.286·25-s + 1.31·26-s + 0.745·27-s − 0.567·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{3} T \) |
| 11 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + 19 T + p^{7} T^{2} \) |
| 5 | \( 1 - 317 T + p^{7} T^{2} \) |
| 7 | \( 1 + 1030 T + p^{7} T^{2} \) |
| 13 | \( 1 + 14676 T + p^{7} T^{2} \) |
| 17 | \( 1 + 30058 T + p^{7} T^{2} \) |
| 19 | \( 1 - 38056 T + p^{7} T^{2} \) |
| 23 | \( 1 + 12911 T + p^{7} T^{2} \) |
| 29 | \( 1 + 3120 p T + p^{7} T^{2} \) |
| 31 | \( 1 + 139023 T + p^{7} T^{2} \) |
| 37 | \( 1 - 251511 T + p^{7} T^{2} \) |
| 41 | \( 1 + 318192 T + p^{7} T^{2} \) |
| 43 | \( 1 - 672430 T + p^{7} T^{2} \) |
| 47 | \( 1 + 519096 T + p^{7} T^{2} \) |
| 53 | \( 1 - 773570 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2194167 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3163180 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1293557 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1207245 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4724772 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2638102 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4830962 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2448233 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3948601 T + p^{7} T^{2} \) |
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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
−16.21637812023542635851984419446, −14.53637051457676112105041844620, −13.07266250617324328498757170488, −11.61379274514151473684799559344, −9.984057296592824883496852982477, −9.206861503295594585133351494411, −6.97088645238078387965522020762, −5.63761719544285357490124811128, −2.49150011716777123923808972469, 0,
2.49150011716777123923808972469, 5.63761719544285357490124811128, 6.97088645238078387965522020762, 9.206861503295594585133351494411, 9.984057296592824883496852982477, 11.61379274514151473684799559344, 13.07266250617324328498757170488, 14.53637051457676112105041844620, 16.21637812023542635851984419446
