| L(s) = 1 | − 117·2-s + 9.59e3·4-s + 1.56e4·5-s − 8.72e4·7-s − 6.43e5·8-s + 5.31e5·9-s − 1.82e6·10-s + 1.77e6·11-s + 9.62e6·13-s + 1.02e7·14-s + 3.59e7·16-s + 3.57e7·17-s − 6.21e7·18-s + 1.49e8·20-s − 2.07e8·22-s + 2.44e8·25-s − 1.12e9·26-s − 8.37e8·28-s + 3.46e8·31-s − 1.57e9·32-s − 4.18e9·34-s − 1.36e9·35-s + 5.09e9·36-s − 1.00e10·40-s − 7.56e9·43-s + 1.69e10·44-s + 8.30e9·45-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.34·4-s + 5-s − 0.741·7-s − 2.45·8-s + 9-s − 1.82·10-s + 11-s + 1.99·13-s + 1.35·14-s + 2.14·16-s + 1.48·17-s − 1.82·18-s + 2.34·20-s − 1.82·22-s + 25-s − 3.64·26-s − 1.73·28-s + 0.390·31-s − 1.46·32-s − 2.70·34-s − 0.741·35-s + 2.34·36-s − 2.45·40-s − 1.19·43-s + 2.34·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.425996607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425996607\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{6} T \) |
| 11 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 + 117 T + p^{12} T^{2} \) |
| 3 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( 1 + 87282 T + p^{12} T^{2} \) |
| 13 | \( 1 - 9629118 T + p^{12} T^{2} \) |
| 17 | \( 1 - 35761518 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 31 | \( 1 - 346700482 T + p^{12} T^{2} \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 43 | \( 1 + 7565541282 T + p^{12} T^{2} \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 84345242798 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( 1 - 240883653922 T + p^{12} T^{2} \) |
| 73 | \( 1 - 297569550798 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( 1 + 579981676482 T + p^{12} T^{2} \) |
| 89 | \( 1 + 991704621598 T + p^{12} T^{2} \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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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
−12.49508958983650476494352722986, −11.05308719070058572795757337376, −9.994663002935162214430857478235, −9.424088083440271636699166552526, −8.305514843770271162440566970127, −6.80858396647608582180258975616, −6.08946626516569082083404110778, −3.37615767557314882074441038295, −1.58286685038260409255782335399, −1.01463658497175101407862335232,
1.01463658497175101407862335232, 1.58286685038260409255782335399, 3.37615767557314882074441038295, 6.08946626516569082083404110778, 6.80858396647608582180258975616, 8.305514843770271162440566970127, 9.424088083440271636699166552526, 9.994663002935162214430857478235, 11.05308719070058572795757337376, 12.49508958983650476494352722986
