| L(s) = 1 | + 32·2-s − 243·3-s + 1.02e3·4-s − 3.12e3·5-s − 7.77e3·6-s + 2.93e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 1.00e5·10-s − 5.38e5·11-s − 2.48e5·12-s − 2.56e4·13-s + 9.39e5·14-s + 7.59e5·15-s + 1.04e6·16-s − 9.80e6·17-s + 1.88e6·18-s − 6.59e6·19-s − 3.20e6·20-s − 7.13e6·21-s − 1.72e7·22-s − 3.50e7·23-s − 7.96e6·24-s + 9.76e6·25-s − 8.19e5·26-s − 1.43e7·27-s + 3.00e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.659·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.00·11-s − 0.288·12-s − 0.0191·13-s + 0.466·14-s + 0.258·15-s + 1/4·16-s − 1.67·17-s + 0.235·18-s − 0.611·19-s − 0.223·20-s − 0.381·21-s − 0.713·22-s − 1.13·23-s − 0.204·24-s + 1/5·25-s − 0.0135·26-s − 0.192·27-s + 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 7 | \( 1 - 29348 T + p^{11} T^{2} \) |
| 11 | \( 1 + 538680 T + p^{11} T^{2} \) |
| 13 | \( 1 + 25606 T + p^{11} T^{2} \) |
| 17 | \( 1 + 9807162 T + p^{11} T^{2} \) |
| 19 | \( 1 + 6599596 T + p^{11} T^{2} \) |
| 23 | \( 1 + 35008200 T + p^{11} T^{2} \) |
| 29 | \( 1 - 1288146 T + p^{11} T^{2} \) |
| 31 | \( 1 + 50347432 T + p^{11} T^{2} \) |
| 37 | \( 1 + 650167894 T + p^{11} T^{2} \) |
| 41 | \( 1 + 678700566 T + p^{11} T^{2} \) |
| 43 | \( 1 - 354979292 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1215951480 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4566555138 T + p^{11} T^{2} \) |
| 59 | \( 1 + 2196450120 T + p^{11} T^{2} \) |
| 61 | \( 1 - 9925999550 T + p^{11} T^{2} \) |
| 67 | \( 1 + 674495812 T + p^{11} T^{2} \) |
| 71 | \( 1 - 17538228960 T + p^{11} T^{2} \) |
| 73 | \( 1 - 19619940914 T + p^{11} T^{2} \) |
| 79 | \( 1 + 1369906648 T + p^{11} T^{2} \) |
| 83 | \( 1 + 62181646116 T + p^{11} T^{2} \) |
| 89 | \( 1 + 64990633758 T + p^{11} T^{2} \) |
| 97 | \( 1 - 101104524386 T + p^{11} T^{2} \) |
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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
−13.75268921679793465417842795825, −12.58977457693232611716509065747, −11.41798736922253773824459425411, −10.49803026810091164170624032287, −8.330296051615115047621196345566, −6.87935729082871977325106614400, −5.32721241934688003536160564471, −4.15131361169759258540840191467, −2.12853351324170830449451015129, 0,
2.12853351324170830449451015129, 4.15131361169759258540840191467, 5.32721241934688003536160564471, 6.87935729082871977325106614400, 8.330296051615115047621196345566, 10.49803026810091164170624032287, 11.41798736922253773824459425411, 12.58977457693232611716509065747, 13.75268921679793465417842795825
