| L(s) = 1 | − 1.12·2-s − 27·3-s − 126.·4-s − 79.7·5-s + 30.5·6-s + 71.6·7-s + 287.·8-s + 729·9-s + 90.0·10-s + 3.46e3·11-s + 3.42e3·12-s + 6.67e3·13-s − 80.9·14-s + 2.15e3·15-s + 1.58e4·16-s − 7.50e3·17-s − 823.·18-s + 2.23e4·19-s + 1.01e4·20-s − 1.93e3·21-s − 3.91e3·22-s + 9.09e4·23-s − 7.76e3·24-s − 7.17e4·25-s − 7.53e3·26-s − 1.96e4·27-s − 9.08e3·28-s + ⋯ |
| L(s) = 1 | − 0.0998·2-s − 0.577·3-s − 0.990·4-s − 0.285·5-s + 0.0576·6-s + 0.0790·7-s + 0.198·8-s + 0.333·9-s + 0.0284·10-s + 0.784·11-s + 0.571·12-s + 0.842·13-s − 0.00788·14-s + 0.164·15-s + 0.970·16-s − 0.370·17-s − 0.0332·18-s + 0.749·19-s + 0.282·20-s − 0.0456·21-s − 0.0783·22-s + 1.55·23-s − 0.114·24-s − 0.918·25-s − 0.0841·26-s − 0.192·27-s − 0.0782·28-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
| \(L(4)\) |
\(\approx\) |
\(1.310919458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310919458\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 1.12T + 128T^{2} \) |
| 5 | \( 1 + 79.7T + 7.81e4T^{2} \) |
| 7 | \( 1 - 71.6T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.46e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.50e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.86e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.07e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.80e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.33e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.76e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.53e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.53e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.86e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.14e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.47e6T + 8.07e13T^{2} \) |
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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
−9.729825405196173271942040337254, −9.050233258759920653060926392364, −8.178238971300637520936370863834, −7.12195185623146942601515961179, −6.07254905201293109260893612844, −5.10222674033773489324474416085, −4.21052594653654265275576331153, −3.32121872617276651438007312507, −1.43968171753697867292160057491, −0.59849270788633538446828333360,
0.59849270788633538446828333360, 1.43968171753697867292160057491, 3.32121872617276651438007312507, 4.21052594653654265275576331153, 5.10222674033773489324474416085, 6.07254905201293109260893612844, 7.12195185623146942601515961179, 8.178238971300637520936370863834, 9.050233258759920653060926392364, 9.729825405196173271942040337254
