| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 60·5-s − 36·6-s − 112·7-s − 64·8-s + 81·9-s + 240·10-s + 336·11-s + 144·12-s + 1.10e3·13-s + 448·14-s − 540·15-s + 256·16-s + 192·17-s − 324·18-s − 952·19-s − 960·20-s − 1.00e3·21-s − 1.34e3·22-s + 114·23-s − 576·24-s + 475·25-s − 4.42e3·26-s + 729·27-s − 1.79e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.07·5-s − 0.408·6-s − 0.863·7-s − 0.353·8-s + 1/3·9-s + 0.758·10-s + 0.837·11-s + 0.288·12-s + 1.81·13-s + 0.610·14-s − 0.619·15-s + 1/4·16-s + 0.161·17-s − 0.235·18-s − 0.604·19-s − 0.536·20-s − 0.498·21-s − 0.592·22-s + 0.0449·23-s − 0.204·24-s + 0.151·25-s − 1.28·26-s + 0.192·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 37 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 12 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 16 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 336 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 - 192 T + p^{5} T^{2} \) |
| 19 | \( 1 + 952 T + p^{5} T^{2} \) |
| 23 | \( 1 - 114 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8484 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2024 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15630 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3748 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1176 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4686 T + p^{5} T^{2} \) |
| 59 | \( 1 + 26370 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14342 T + p^{5} T^{2} \) |
| 67 | \( 1 + 11188 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31080 T + p^{5} T^{2} \) |
| 73 | \( 1 - 45542 T + p^{5} T^{2} \) |
| 79 | \( 1 + 45796 T + p^{5} T^{2} \) |
| 83 | \( 1 + 76296 T + p^{5} T^{2} \) |
| 89 | \( 1 + 4308 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119650 T + p^{5} T^{2} \) |
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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
−10.93570414150473709801482097906, −9.735302296722282817870696508620, −8.824470955187786271109821555059, −8.140958237696093851533719517437, −7.02272740202135155258874309468, −6.09821824040773799789126233995, −3.96923920950967712346036583392, −3.32405914490831877835528874391, −1.49064631200681284383839589575, 0,
1.49064631200681284383839589575, 3.32405914490831877835528874391, 3.96923920950967712346036583392, 6.09821824040773799789126233995, 7.02272740202135155258874309468, 8.140958237696093851533719517437, 8.824470955187786271109821555059, 9.735302296722282817870696508620, 10.93570414150473709801482097906
