| L(s) = 1 | + 6.57·2-s − 9·3-s + 11.1·4-s − 5.02·5-s − 59.1·6-s − 195.·7-s − 136.·8-s + 81·9-s − 33.0·10-s + 615.·11-s − 100.·12-s + 601.·13-s − 1.28e3·14-s + 45.2·15-s − 1.25e3·16-s + 1.58e3·17-s + 532.·18-s + 1.49e3·19-s − 56.1·20-s + 1.75e3·21-s + 4.04e3·22-s + 4.62e3·23-s + 1.23e3·24-s − 3.09e3·25-s + 3.95e3·26-s − 729·27-s − 2.17e3·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.349·4-s − 0.0899·5-s − 0.670·6-s − 1.50·7-s − 0.756·8-s + 0.333·9-s − 0.104·10-s + 1.53·11-s − 0.201·12-s + 0.987·13-s − 1.74·14-s + 0.0519·15-s − 1.22·16-s + 1.32·17-s + 0.387·18-s + 0.950·19-s − 0.0314·20-s + 0.869·21-s + 1.78·22-s + 1.82·23-s + 0.436·24-s − 0.991·25-s + 1.14·26-s − 0.192·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.469892225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469892225\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 6.57T + 32T^{2} \) |
| 5 | \( 1 + 5.02T + 3.12e3T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 615.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 601.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.76e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.04e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.51e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.38e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.40e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.27e5T + 8.58e9T^{2} \) |
| 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
−12.02303234944216505872569678426, −11.23439098863289923626589762625, −9.703534514504497068190910975451, −9.078956734254250776437510370002, −7.11469873902245674170245906602, −6.19686194787065571315958071257, −5.46700608027007486391142232175, −3.87343548471956244243686942622, −3.30926351532551741667386799788, −0.893729549963620601188906258580,
0.893729549963620601188906258580, 3.30926351532551741667386799788, 3.87343548471956244243686942622, 5.46700608027007486391142232175, 6.19686194787065571315958071257, 7.11469873902245674170245906602, 9.078956734254250776437510370002, 9.703534514504497068190910975451, 11.23439098863289923626589762625, 12.02303234944216505872569678426
