| L(s) = 1 | − 64·2-s + 1.22e3·3-s + 4.09e3·4-s + 1.56e4·5-s − 7.83e4·6-s − 6.52e4·7-s − 2.62e5·8-s − 9.61e4·9-s − 1.00e6·10-s + 7.42e6·11-s + 5.01e6·12-s + 3.22e7·13-s + 4.17e6·14-s + 1.91e7·15-s + 1.67e7·16-s − 2.00e7·17-s + 6.15e6·18-s + 7.70e7·19-s + 6.40e7·20-s − 7.98e7·21-s − 4.75e8·22-s + 6.64e8·23-s − 3.20e8·24-s + 2.44e8·25-s − 2.06e9·26-s − 2.06e9·27-s − 2.67e8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.969·3-s + 1/2·4-s + 0.447·5-s − 0.685·6-s − 0.209·7-s − 0.353·8-s − 0.0603·9-s − 0.316·10-s + 1.26·11-s + 0.484·12-s + 1.85·13-s + 0.148·14-s + 0.433·15-s + 1/4·16-s − 0.201·17-s + 0.0426·18-s + 0.375·19-s + 0.223·20-s − 0.203·21-s − 0.893·22-s + 0.935·23-s − 0.342·24-s + 1/5·25-s − 1.31·26-s − 1.02·27-s − 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.968270391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968270391\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{6} T \) |
| 5 | \( 1 - p^{6} T \) |
| good | 3 | \( 1 - 136 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 + 9316 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 7427652 T + p^{13} T^{2} \) |
| 13 | \( 1 - 32243054 T + p^{13} T^{2} \) |
| 17 | \( 1 + 20088222 T + p^{13} T^{2} \) |
| 19 | \( 1 - 77070740 T + p^{13} T^{2} \) |
| 23 | \( 1 - 664071804 T + p^{13} T^{2} \) |
| 29 | \( 1 - 1558250670 T + p^{13} T^{2} \) |
| 31 | \( 1 + 303290968 T + p^{13} T^{2} \) |
| 37 | \( 1 + 775029322 T + p^{13} T^{2} \) |
| 41 | \( 1 - 43696205082 T + p^{13} T^{2} \) |
| 43 | \( 1 + 68680553536 T + p^{13} T^{2} \) |
| 47 | \( 1 + 138979393812 T + p^{13} T^{2} \) |
| 53 | \( 1 + 103656826986 T + p^{13} T^{2} \) |
| 59 | \( 1 - 394887188940 T + p^{13} T^{2} \) |
| 61 | \( 1 + 488570895538 T + p^{13} T^{2} \) |
| 67 | \( 1 - 368381730848 T + p^{13} T^{2} \) |
| 71 | \( 1 - 325473704592 T + p^{13} T^{2} \) |
| 73 | \( 1 + 2262556998406 T + p^{13} T^{2} \) |
| 79 | \( 1 - 2032917332000 T + p^{13} T^{2} \) |
| 83 | \( 1 + 854518199496 T + p^{13} T^{2} \) |
| 89 | \( 1 - 8906829484890 T + p^{13} T^{2} \) |
| 97 | \( 1 + 9873550533742 T + p^{13} T^{2} \) |
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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
−17.72599994620068940891037464557, −16.26912573741230595391175146798, −14.69149720199710449028643896768, −13.41146741940031858626341219347, −11.27681103912710926282584651644, −9.409019812515722734204273411400, −8.446453898784722073523927146943, −6.41348796693541393420607706456, −3.32764419861210783395018957960, −1.40653750997997567616527119429,
1.40653750997997567616527119429, 3.32764419861210783395018957960, 6.41348796693541393420607706456, 8.446453898784722073523927146943, 9.409019812515722734204273411400, 11.27681103912710926282584651644, 13.41146741940031858626341219347, 14.69149720199710449028643896768, 16.26912573741230595391175146798, 17.72599994620068940891037464557
