| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 216·6-s + 1.42e3·7-s + 512·8-s + 729·9-s + 5.07e3·11-s − 1.72e3·12-s − 1.18e4·13-s + 1.14e4·14-s + 4.09e3·16-s + 1.92e4·17-s + 5.83e3·18-s − 4.37e4·19-s − 3.85e4·21-s + 4.05e4·22-s − 3.15e3·23-s − 1.38e4·24-s − 9.51e4·26-s − 1.96e4·27-s + 9.13e4·28-s + 1.40e5·29-s + 1.47e5·31-s + 3.27e4·32-s − 1.36e5·33-s + 1.53e5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.57·7-s + 0.353·8-s + 1/3·9-s + 1.14·11-s − 0.288·12-s − 1.50·13-s + 1.11·14-s + 1/4·16-s + 0.949·17-s + 0.235·18-s − 1.46·19-s − 0.907·21-s + 0.812·22-s − 0.0539·23-s − 0.204·24-s − 1.06·26-s − 0.192·27-s + 0.786·28-s + 1.06·29-s + 0.889·31-s + 0.176·32-s − 0.663·33-s + 0.671·34-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
| \(L(4)\) |
\(\approx\) |
\(3.433640425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.433640425\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 1427 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5070 T + p^{7} T^{2} \) |
| 13 | \( 1 + 11899 T + p^{7} T^{2} \) |
| 17 | \( 1 - 19242 T + p^{7} T^{2} \) |
| 19 | \( 1 + 43711 T + p^{7} T^{2} \) |
| 23 | \( 1 + 3150 T + p^{7} T^{2} \) |
| 29 | \( 1 - 140106 T + p^{7} T^{2} \) |
| 31 | \( 1 - 147563 T + p^{7} T^{2} \) |
| 37 | \( 1 - 561674 T + p^{7} T^{2} \) |
| 41 | \( 1 + 270336 T + p^{7} T^{2} \) |
| 43 | \( 1 - 180683 T + p^{7} T^{2} \) |
| 47 | \( 1 - 97470 T + p^{7} T^{2} \) |
| 53 | \( 1 - 2130132 T + p^{7} T^{2} \) |
| 59 | \( 1 + 935070 T + p^{7} T^{2} \) |
| 61 | \( 1 - 135875 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1443433 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2685840 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3280466 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5672672 T + p^{7} T^{2} \) |
| 83 | \( 1 - 4227906 T + p^{7} T^{2} \) |
| 89 | \( 1 + 11890848 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3808529 T + p^{7} T^{2} \) |
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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
−11.91782107576065407980410612910, −10.96267370681316540787265873548, −9.914281959921017513499387874198, −8.362666064396318072497082946062, −7.29698206031884606057449777316, −6.13345171564187363459580087431, −4.90356798217124722183132470678, −4.24994153770749011432153715931, −2.31954374933200002679610544640, −1.05049753193279271650008182459,
1.05049753193279271650008182459, 2.31954374933200002679610544640, 4.24994153770749011432153715931, 4.90356798217124722183132470678, 6.13345171564187363459580087431, 7.29698206031884606057449777316, 8.362666064396318072497082946062, 9.914281959921017513499387874198, 10.96267370681316540787265873548, 11.91782107576065407980410612910
