| L(s) = 1 | + 4·2-s − 29·3-s + 16·4-s − 31·5-s − 116·6-s − 230·7-s + 64·8-s + 598·9-s − 124·10-s + 121·11-s − 464·12-s + 112·13-s − 920·14-s + 899·15-s + 256·16-s − 1.14e3·17-s + 2.39e3·18-s − 612·19-s − 496·20-s + 6.67e3·21-s + 484·22-s − 1.94e3·23-s − 1.85e3·24-s − 2.16e3·25-s + 448·26-s − 1.02e4·27-s − 3.68e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 1/2·4-s − 0.554·5-s − 1.31·6-s − 1.77·7-s + 0.353·8-s + 2.46·9-s − 0.392·10-s + 0.301·11-s − 0.930·12-s + 0.183·13-s − 1.25·14-s + 1.03·15-s + 1/4·16-s − 0.958·17-s + 1.74·18-s − 0.388·19-s − 0.277·20-s + 3.30·21-s + 0.213·22-s − 0.765·23-s − 0.657·24-s − 0.692·25-s + 0.129·26-s − 2.71·27-s − 0.887·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 29 T + p^{5} T^{2} \) |
| 5 | \( 1 + 31 T + p^{5} T^{2} \) |
| 7 | \( 1 + 230 T + p^{5} T^{2} \) |
| 13 | \( 1 - 112 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1142 T + p^{5} T^{2} \) |
| 19 | \( 1 + 612 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1941 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1192 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1037 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8083 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10444 T + p^{5} T^{2} \) |
| 43 | \( 1 - 58 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8656 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20318 T + p^{5} T^{2} \) |
| 59 | \( 1 + 21351 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47044 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48093 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24967 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42288 T + p^{5} T^{2} \) |
| 79 | \( 1 + 72410 T + p^{5} T^{2} \) |
| 83 | \( 1 + 15806 T + p^{5} T^{2} \) |
| 89 | \( 1 + 114761 T + p^{5} T^{2} \) |
| 97 | \( 1 + 5159 T + p^{5} T^{2} \) |
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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
−16.15595010129234074381616164301, −15.61923593718955118297052990975, −13.21440476609437154624754622737, −12.33297780937294046829455662146, −11.29888320733400927834504572776, −9.976571960152509978248600298334, −6.84593743536970044315079720845, −5.97424717232060262937512525245, −4.10995632237962299247234454324, 0,
4.10995632237962299247234454324, 5.97424717232060262937512525245, 6.84593743536970044315079720845, 9.976571960152509978248600298334, 11.29888320733400927834504572776, 12.33297780937294046829455662146, 13.21440476609437154624754622737, 15.61923593718955118297052990975, 16.15595010129234074381616164301
