| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 125·5-s + 216·6-s − 1.15e3·7-s + 512·8-s + 729·9-s + 1.00e3·10-s + 1.69e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 9.23e3·14-s + 3.37e3·15-s + 4.09e3·16-s + 5.11e3·17-s + 5.83e3·18-s − 3.91e4·19-s + 8.00e3·20-s − 3.11e4·21-s + 1.35e4·22-s − 4.00e4·23-s + 1.38e4·24-s + 1.56e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 7.38e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.27·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.383·11-s + 0.288·12-s − 0.277·13-s − 0.899·14-s + 0.258·15-s + 1/4·16-s + 0.252·17-s + 0.235·18-s − 1.30·19-s + 0.223·20-s − 0.734·21-s + 0.271·22-s − 0.686·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| 13 | \( 1 + p^{3} T \) |
| good | 7 | \( 1 + 1154 T + p^{7} T^{2} \) |
| 11 | \( 1 - 1692 T + p^{7} T^{2} \) |
| 17 | \( 1 - 5116 T + p^{7} T^{2} \) |
| 19 | \( 1 + 39146 T + p^{7} T^{2} \) |
| 23 | \( 1 + 40030 T + p^{7} T^{2} \) |
| 29 | \( 1 + 32224 T + p^{7} T^{2} \) |
| 31 | \( 1 + 12572 T + p^{7} T^{2} \) |
| 37 | \( 1 - 261998 T + p^{7} T^{2} \) |
| 41 | \( 1 - 176154 T + p^{7} T^{2} \) |
| 43 | \( 1 + 733212 T + p^{7} T^{2} \) |
| 47 | \( 1 + 645864 T + p^{7} T^{2} \) |
| 53 | \( 1 + 794306 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1203300 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1113994 T + p^{7} T^{2} \) |
| 67 | \( 1 + 950728 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4666384 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1512984 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2913448 T + p^{7} T^{2} \) |
| 83 | \( 1 + 199996 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4236506 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6845376 T + p^{7} T^{2} \) |
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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
−9.749100127845076421899240029166, −8.867106469370261001920109231032, −7.69186581082838044028610180452, −6.55845561840442711299878486016, −6.04921126344695044470157845636, −4.63984625869494384664698764860, −3.61649348593105664964258293027, −2.73844113527284410270123718214, −1.67411468505396470995718771769, 0,
1.67411468505396470995718771769, 2.73844113527284410270123718214, 3.61649348593105664964258293027, 4.63984625869494384664698764860, 6.04921126344695044470157845636, 6.55845561840442711299878486016, 7.69186581082838044028610180452, 8.867106469370261001920109231032, 9.749100127845076421899240029166
