| L(s) = 1 | − 78·2-s + 4.03e3·4-s + 5.37e3·5-s − 2.77e4·7-s − 1.55e5·8-s − 4.18e5·10-s − 6.37e5·11-s + 7.66e5·13-s + 2.16e6·14-s + 3.82e6·16-s − 3.08e6·17-s − 1.95e7·19-s + 2.16e7·20-s + 4.97e7·22-s − 1.53e7·23-s − 1.99e7·25-s − 5.97e7·26-s − 1.12e8·28-s − 1.07e7·29-s − 5.09e7·31-s + 1.88e7·32-s + 2.40e8·34-s − 1.49e8·35-s + 6.64e8·37-s + 1.52e9·38-s − 8.32e8·40-s − 8.98e8·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.97·4-s + 0.768·5-s − 0.624·7-s − 1.67·8-s − 1.32·10-s − 1.19·11-s + 0.572·13-s + 1.07·14-s + 0.912·16-s − 0.526·17-s − 1.80·19-s + 1.51·20-s + 2.05·22-s − 0.496·23-s − 0.409·25-s − 0.986·26-s − 1.23·28-s − 0.0973·29-s − 0.319·31-s + 0.0995·32-s + 0.908·34-s − 0.479·35-s + 1.57·37-s + 3.11·38-s − 1.28·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 39 p T + p^{11} T^{2} \) |
| 5 | \( 1 - 1074 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 27760 T + p^{11} T^{2} \) |
| 11 | \( 1 + 637836 T + p^{11} T^{2} \) |
| 13 | \( 1 - 766214 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3084354 T + p^{11} T^{2} \) |
| 19 | \( 1 + 1026916 p T + p^{11} T^{2} \) |
| 23 | \( 1 + 15312360 T + p^{11} T^{2} \) |
| 29 | \( 1 + 10751262 T + p^{11} T^{2} \) |
| 31 | \( 1 + 50937400 T + p^{11} T^{2} \) |
| 37 | \( 1 - 664740830 T + p^{11} T^{2} \) |
| 41 | \( 1 + 898833450 T + p^{11} T^{2} \) |
| 43 | \( 1 + 957947188 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1555741344 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3792417030 T + p^{11} T^{2} \) |
| 59 | \( 1 + 555306924 T + p^{11} T^{2} \) |
| 61 | \( 1 - 4950420998 T + p^{11} T^{2} \) |
| 67 | \( 1 - 5292399284 T + p^{11} T^{2} \) |
| 71 | \( 1 - 14831086248 T + p^{11} T^{2} \) |
| 73 | \( 1 - 13971005210 T + p^{11} T^{2} \) |
| 79 | \( 1 - 3720542360 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8768454036 T + p^{11} T^{2} \) |
| 89 | \( 1 - 25472769174 T + p^{11} T^{2} \) |
| 97 | \( 1 + 39092494846 T + p^{11} T^{2} \) |
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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
−17.96916981872020931219858876825, −16.79222500352226814965278233121, −15.54493925242647643105644211134, −13.13918486900932676128407649521, −10.84355412425302559310099702349, −9.736783513324771300245756503773, −8.271273928381884312057470835274, −6.37766442905269148354389897553, −2.15458349121467029551547150035, 0,
2.15458349121467029551547150035, 6.37766442905269148354389897553, 8.271273928381884312057470835274, 9.736783513324771300245756503773, 10.84355412425302559310099702349, 13.13918486900932676128407649521, 15.54493925242647643105644211134, 16.79222500352226814965278233121, 17.96916981872020931219858876825
