| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 216·6-s + 1.57e3·7-s − 512·8-s + 729·9-s + 7.33e3·11-s − 1.72e3·12-s + 3.80e3·13-s − 1.26e4·14-s + 4.09e3·16-s + 6.60e3·17-s − 5.83e3·18-s + 2.48e4·19-s − 4.25e4·21-s − 5.86e4·22-s − 4.14e4·23-s + 1.38e4·24-s − 3.04e4·26-s − 1.96e4·27-s + 1.00e5·28-s − 4.16e4·29-s + 3.31e4·31-s − 3.27e4·32-s − 1.97e5·33-s − 5.28e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.73·7-s − 0.353·8-s + 1/3·9-s + 1.66·11-s − 0.288·12-s + 0.479·13-s − 1.22·14-s + 1/4·16-s + 0.326·17-s − 0.235·18-s + 0.831·19-s − 1.00·21-s − 1.17·22-s − 0.710·23-s + 0.204·24-s − 0.339·26-s − 0.192·27-s + 0.868·28-s − 0.316·29-s + 0.199·31-s − 0.176·32-s − 0.958·33-s − 0.230·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.826741219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826741219\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 1576 T + p^{7} T^{2} \) |
| 11 | \( 1 - 7332 T + p^{7} T^{2} \) |
| 13 | \( 1 - 3802 T + p^{7} T^{2} \) |
| 17 | \( 1 - 6606 T + p^{7} T^{2} \) |
| 19 | \( 1 - 24860 T + p^{7} T^{2} \) |
| 23 | \( 1 + 41448 T + p^{7} T^{2} \) |
| 29 | \( 1 + 41610 T + p^{7} T^{2} \) |
| 31 | \( 1 - 33152 T + p^{7} T^{2} \) |
| 37 | \( 1 - 36466 T + p^{7} T^{2} \) |
| 41 | \( 1 + 639078 T + p^{7} T^{2} \) |
| 43 | \( 1 - 156412 T + p^{7} T^{2} \) |
| 47 | \( 1 - 433776 T + p^{7} T^{2} \) |
| 53 | \( 1 + 786078 T + p^{7} T^{2} \) |
| 59 | \( 1 - 745140 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1660618 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3290836 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5716152 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2659898 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3807440 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2229468 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5991210 T + p^{7} T^{2} \) |
| 97 | \( 1 - 4060126 T + p^{7} T^{2} \) |
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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
−11.59658129176378751025832469825, −10.84246462691176105711310791503, −9.616503992117380022083344449151, −8.552469027625758417914926685701, −7.60690251350437990709288380429, −6.42877251624005901769994287122, −5.20471875629606065454527492274, −3.89563442687369930324964820199, −1.75073642595848909369178768169, −0.985896580841533879795784573004,
0.985896580841533879795784573004, 1.75073642595848909369178768169, 3.89563442687369930324964820199, 5.20471875629606065454527492274, 6.42877251624005901769994287122, 7.60690251350437990709288380429, 8.552469027625758417914926685701, 9.616503992117380022083344449151, 10.84246462691176105711310791503, 11.59658129176378751025832469825
