| L(s) = 1 | + 16.2·2-s + 134.·4-s − 395.·5-s + 1.52e3·7-s + 105.·8-s − 6.41e3·10-s + 977.·11-s + 7.62e3·13-s + 2.47e4·14-s − 1.55e4·16-s − 1.35e4·17-s + 25.2·19-s − 5.32e4·20-s + 1.58e4·22-s + 1.21e4·23-s + 7.86e4·25-s + 1.23e5·26-s + 2.05e5·28-s + 9.68e4·29-s + 2.66e5·31-s − 2.64e5·32-s − 2.19e5·34-s − 6.04e5·35-s + 3.98e4·37-s + 408.·38-s − 4.17e4·40-s + 7.46e5·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.05·4-s − 1.41·5-s + 1.68·7-s + 0.0727·8-s − 2.02·10-s + 0.221·11-s + 0.962·13-s + 2.41·14-s − 0.946·16-s − 0.668·17-s + 0.000843·19-s − 1.48·20-s + 0.316·22-s + 0.208·23-s + 1.00·25-s + 1.37·26-s + 1.76·28-s + 0.737·29-s + 1.60·31-s − 1.42·32-s − 0.957·34-s − 2.38·35-s + 0.129·37-s + 0.00120·38-s − 0.103·40-s + 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.519263026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.519263026\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 16.2T + 128T^{2} \) |
| 5 | \( 1 + 395.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.52e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 977.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.62e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.35e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 25.2T + 8.93e8T^{2} \) |
| 29 | \( 1 - 9.68e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.66e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.98e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.79e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.49e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.55e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.35e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.13e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.47e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.63e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.65e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.49e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.01e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.22e7T + 8.07e13T^{2} \) |
| 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
−11.50521648257757957433048111408, −10.81443320691185667258705273102, −8.747796058246127961533089633901, −8.046627610305204701482980479749, −6.90796950928345551915632702965, −5.60053250701318947795530672334, −4.36984903070116974308475122722, −4.12818062558959863833328787017, −2.62784412247754257516816300939, −0.955984446695955002211508736101,
0.955984446695955002211508736101, 2.62784412247754257516816300939, 4.12818062558959863833328787017, 4.36984903070116974308475122722, 5.60053250701318947795530672334, 6.90796950928345551915632702965, 8.046627610305204701482980479749, 8.747796058246127961533089633901, 10.81443320691185667258705273102, 11.50521648257757957433048111408
