| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 36·6-s − 32·7-s − 64·8-s + 81·9-s + 12·11-s − 144·12-s + 154·13-s + 128·14-s + 256·16-s + 918·17-s − 324·18-s − 1.06e3·19-s + 288·21-s − 48·22-s + 4.22e3·23-s + 576·24-s − 616·26-s − 729·27-s − 512·28-s − 7.89e3·29-s + 5.19e3·31-s − 1.02e3·32-s − 108·33-s − 3.67e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.246·7-s − 0.353·8-s + 1/3·9-s + 0.0299·11-s − 0.288·12-s + 0.252·13-s + 0.174·14-s + 1/4·16-s + 0.770·17-s − 0.235·18-s − 0.673·19-s + 0.142·21-s − 0.0211·22-s + 1.66·23-s + 0.204·24-s − 0.178·26-s − 0.192·27-s − 0.123·28-s − 1.74·29-s + 0.970·31-s − 0.176·32-s − 0.0172·33-s − 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 32 T + p^{5} T^{2} \) |
| 11 | \( 1 - 12 T + p^{5} T^{2} \) |
| 13 | \( 1 - 154 T + p^{5} T^{2} \) |
| 17 | \( 1 - 54 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4224 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7890 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5192 T + p^{5} T^{2} \) |
| 37 | \( 1 + 16382 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3642 T + p^{5} T^{2} \) |
| 43 | \( 1 + 15116 T + p^{5} T^{2} \) |
| 47 | \( 1 + 23592 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16074 T + p^{5} T^{2} \) |
| 59 | \( 1 + 14340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 47938 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33092 T + p^{5} T^{2} \) |
| 71 | \( 1 - 51912 T + p^{5} T^{2} \) |
| 73 | \( 1 + 12026 T + p^{5} T^{2} \) |
| 79 | \( 1 - 25160 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35796 T + p^{5} T^{2} \) |
| 89 | \( 1 + 75510 T + p^{5} T^{2} \) |
| 97 | \( 1 - 44158 T + p^{5} T^{2} \) |
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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
−11.40961606603456644274022233220, −10.58346481202721189262039899155, −9.603202310276728478840236212483, −8.543716622689154843240981638294, −7.30327133440497150588107756130, −6.32421855281629119371631708818, −5.07509869100264663889510117264, −3.32779548758304047209365682688, −1.50319698133089939801483204710, 0,
1.50319698133089939801483204710, 3.32779548758304047209365682688, 5.07509869100264663889510117264, 6.32421855281629119371631708818, 7.30327133440497150588107756130, 8.543716622689154843240981638294, 9.603202310276728478840236212483, 10.58346481202721189262039899155, 11.40961606603456644274022233220
