| L(s) = 1 | − 64·2-s − 1.83e3·3-s + 4.09e3·4-s + 3.99e3·5-s + 1.17e5·6-s − 4.33e5·7-s − 2.62e5·8-s + 1.77e6·9-s − 2.55e5·10-s + 1.61e6·11-s − 7.52e6·12-s − 1.08e7·13-s + 2.77e7·14-s − 7.32e6·15-s + 1.67e7·16-s + 6.05e7·17-s − 1.13e8·18-s − 2.43e8·19-s + 1.63e7·20-s + 7.95e8·21-s − 1.03e8·22-s − 6.06e8·23-s + 4.81e8·24-s − 1.20e9·25-s + 6.96e8·26-s − 3.34e8·27-s − 1.77e9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.45·3-s + 1/2·4-s + 0.114·5-s + 1.02·6-s − 1.39·7-s − 0.353·8-s + 1.11·9-s − 0.0807·10-s + 0.275·11-s − 0.727·12-s − 0.625·13-s + 0.984·14-s − 0.166·15-s + 1/4·16-s + 0.608·17-s − 0.787·18-s − 1.18·19-s + 0.0571·20-s + 2.02·21-s − 0.194·22-s − 0.853·23-s + 0.514·24-s − 0.986·25-s + 0.441·26-s − 0.166·27-s − 0.696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{6} T \) |
| good | 3 | \( 1 + 68 p^{3} T + p^{13} T^{2} \) |
| 5 | \( 1 - 798 p T + p^{13} T^{2} \) |
| 7 | \( 1 + 433432 T + p^{13} T^{2} \) |
| 11 | \( 1 - 147252 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 10878466 T + p^{13} T^{2} \) |
| 17 | \( 1 - 60569298 T + p^{13} T^{2} \) |
| 19 | \( 1 + 243131740 T + p^{13} T^{2} \) |
| 23 | \( 1 + 606096456 T + p^{13} T^{2} \) |
| 29 | \( 1 - 181332390 p T + p^{13} T^{2} \) |
| 31 | \( 1 + 1824312928 T + p^{13} T^{2} \) |
| 37 | \( 1 + 3005875402 T + p^{13} T^{2} \) |
| 41 | \( 1 + 49704880758 T + p^{13} T^{2} \) |
| 43 | \( 1 - 58766693084 T + p^{13} T^{2} \) |
| 47 | \( 1 + 42095878032 T + p^{13} T^{2} \) |
| 53 | \( 1 + 181140755706 T + p^{13} T^{2} \) |
| 59 | \( 1 - 206730587820 T + p^{13} T^{2} \) |
| 61 | \( 1 + 124479015058 T + p^{13} T^{2} \) |
| 67 | \( 1 - 95665133588 T + p^{13} T^{2} \) |
| 71 | \( 1 + 371436487128 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1800576064726 T + p^{13} T^{2} \) |
| 79 | \( 1 - 1557932091920 T + p^{13} T^{2} \) |
| 83 | \( 1 - 2492790917604 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2994235754490 T + p^{13} T^{2} \) |
| 97 | \( 1 - 4382492665058 T + p^{13} T^{2} \) |
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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
−25.55572663970023698063076155654, −23.43031632637744750084826101922, −21.96660617866132922279426882618, −19.30076164754809625107135652360, −17.42367678302764622274079606362, −16.16691835046758991674984843933, −12.22585173842646883410475762932, −10.12476768920683525916472649759, −6.34565803152834236748804678177, 0,
6.34565803152834236748804678177, 10.12476768920683525916472649759, 12.22585173842646883410475762932, 16.16691835046758991674984843933, 17.42367678302764622274079606362, 19.30076164754809625107135652360, 21.96660617866132922279426882618, 23.43031632637744750084826101922, 25.55572663970023698063076155654
