| L(s) = 1 | − 23·2-s + 273·4-s + 625·5-s + 1.28e3·7-s − 391·8-s + 6.56e3·9-s − 1.43e4·10-s + 1.46e4·11-s − 3.08e4·13-s − 2.94e4·14-s − 6.08e4·16-s − 5.43e3·17-s − 1.50e5·18-s + 1.70e5·20-s − 3.36e5·22-s + 3.90e5·25-s + 7.10e5·26-s + 3.49e5·28-s + 7.06e5·31-s + 1.50e6·32-s + 1.25e5·34-s + 8.01e5·35-s + 1.79e6·36-s − 2.44e5·40-s + 5.56e6·43-s + 3.99e6·44-s + 4.10e6·45-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.06·4-s + 5-s + 0.533·7-s − 0.0954·8-s + 9-s − 1.43·10-s + 11-s − 1.08·13-s − 0.767·14-s − 0.929·16-s − 0.0651·17-s − 1.43·18-s + 1.06·20-s − 1.43·22-s + 25-s + 1.55·26-s + 0.569·28-s + 0.765·31-s + 1.43·32-s + 0.0935·34-s + 0.533·35-s + 1.06·36-s − 0.0954·40-s + 1.62·43-s + 1.06·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.306023219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306023219\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{4} T \) |
| 11 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 + 23 T + p^{8} T^{2} \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 - 1282 T + p^{8} T^{2} \) |
| 13 | \( 1 + 30878 T + p^{8} T^{2} \) |
| 17 | \( 1 + 5438 T + p^{8} T^{2} \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 706562 T + p^{8} T^{2} \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 - 5566882 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( 1 + 12387358 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( 1 + 34839358 T + p^{8} T^{2} \) |
| 73 | \( 1 - 56370562 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 - 90097762 T + p^{8} T^{2} \) |
| 89 | \( 1 - 125357762 T + p^{8} T^{2} \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.66753412069754556491935671653, −12.24400623577549414426302551860, −10.79307235445606406743539395895, −9.801606776071171627755142298209, −9.127689559007839723887304524961, −7.68438297925489974175425333354, −6.57352326877886123066920995327, −4.66480089888989234981502317529, −2.08299614635532685606586800263, −1.03027297756288283171345149524,
1.03027297756288283171345149524, 2.08299614635532685606586800263, 4.66480089888989234981502317529, 6.57352326877886123066920995327, 7.68438297925489974175425333354, 9.127689559007839723887304524961, 9.801606776071171627755142298209, 10.79307235445606406743539395895, 12.24400623577549414426302551860, 13.66753412069754556491935671653
