| L(s) = 1 | + 82·5-s + 456·7-s − 2.52e3·11-s − 1.07e4·13-s + 1.11e4·17-s − 4.12e3·19-s + 8.17e4·23-s − 7.14e4·25-s − 9.97e4·29-s + 4.04e4·31-s + 3.73e4·35-s − 4.19e5·37-s − 1.41e5·41-s + 6.90e5·43-s − 6.82e5·47-s − 6.15e5·49-s − 1.81e6·53-s − 2.06e5·55-s − 9.66e5·59-s + 1.88e6·61-s − 8.83e5·65-s − 2.96e6·67-s − 2.54e6·71-s − 1.68e6·73-s − 1.15e6·77-s − 4.03e6·79-s − 5.38e6·83-s + ⋯ |
| L(s) = 1 | + 0.293·5-s + 0.502·7-s − 0.571·11-s − 1.36·13-s + 0.550·17-s − 0.137·19-s + 1.40·23-s − 0.913·25-s − 0.759·29-s + 0.244·31-s + 0.147·35-s − 1.36·37-s − 0.320·41-s + 1.32·43-s − 0.958·47-s − 0.747·49-s − 1.67·53-s − 0.167·55-s − 0.612·59-s + 1.06·61-s − 0.399·65-s − 1.20·67-s − 0.844·71-s − 0.505·73-s − 0.287·77-s − 0.921·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 82 T + p^{7} T^{2} \) |
| 7 | \( 1 - 456 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 + 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 - 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 - 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 + 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 - 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 + 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 + 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 - 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 + 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 + 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6065758 T + p^{7} T^{2} \) |
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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
−11.24495825251253246524816476571, −10.20007368409000871565066546878, −9.281218954952405929158685121491, −7.984667942244909869666543256060, −7.09536220330353784806406275699, −5.58756303907765568647376765088, −4.68217654750973749859364234940, −2.99392682883034761308568152420, −1.69321747141231145565543321625, 0,
1.69321747141231145565543321625, 2.99392682883034761308568152420, 4.68217654750973749859364234940, 5.58756303907765568647376765088, 7.09536220330353784806406275699, 7.984667942244909869666543256060, 9.281218954952405929158685121491, 10.20007368409000871565066546878, 11.24495825251253246524816476571
