| L(s) = 1 | + 21.1·2-s − 27·3-s + 321.·4-s + 492.·5-s − 572.·6-s − 435.·7-s + 4.09e3·8-s + 729·9-s + 1.04e4·10-s − 214.·11-s − 8.67e3·12-s − 1.22e4·13-s − 9.22e3·14-s − 1.32e4·15-s + 4.56e4·16-s + 2.79e4·17-s + 1.54e4·18-s − 5.30e4·19-s + 1.58e5·20-s + 1.17e4·21-s − 4.55e3·22-s + 1.14e5·23-s − 1.10e5·24-s + 1.64e5·25-s − 2.58e5·26-s − 1.96e4·27-s − 1.39e5·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 0.577·3-s + 2.50·4-s + 1.76·5-s − 1.08·6-s − 0.479·7-s + 2.82·8-s + 0.333·9-s + 3.29·10-s − 0.0486·11-s − 1.44·12-s − 1.54·13-s − 0.898·14-s − 1.01·15-s + 2.78·16-s + 1.37·17-s + 0.624·18-s − 1.77·19-s + 4.41·20-s + 0.276·21-s − 0.0911·22-s + 1.96·23-s − 1.63·24-s + 2.10·25-s − 2.88·26-s − 0.192·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.585124065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.585124065\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 21.1T + 128T^{2} \) |
| 5 | \( 1 - 492.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 435.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 214.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.79e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.30e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.14e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.67e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.87e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.93e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.19e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.76e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.98e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.50e6T + 8.07e13T^{2} \) |
| 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
−10.26069227995871992785491919685, −9.317838715070352737520758483648, −7.44189666358304419255566247770, −6.46476108913226719868260524739, −6.05299223914927671527096348673, −5.10904214102830638046207490698, −4.56883212021700122519352168443, −2.91535628186059863251777722455, −2.38522520727574659411642899104, −1.13226511187527613740627704169,
1.13226511187527613740627704169, 2.38522520727574659411642899104, 2.91535628186059863251777722455, 4.56883212021700122519352168443, 5.10904214102830638046207490698, 6.05299223914927671527096348673, 6.46476108913226719868260524739, 7.44189666358304419255566247770, 9.317838715070352737520758483648, 10.26069227995871992785491919685
