| L(s) = 1 | + 10.9·2-s + 15.7·3-s + 88.2·4-s + 51.6·5-s + 173.·6-s + 617.·8-s + 6.60·9-s + 565.·10-s + 255.·11-s + 1.39e3·12-s − 169·13-s + 815.·15-s + 3.94e3·16-s − 53.2·17-s + 72.4·18-s − 268.·19-s + 4.55e3·20-s + 2.80e3·22-s + 2.08e3·23-s + 9.75e3·24-s − 461.·25-s − 1.85e3·26-s − 3.73e3·27-s − 8.17e3·29-s + 8.94e3·30-s + 4.78e3·31-s + 2.35e4·32-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 1.01·3-s + 2.75·4-s + 0.923·5-s + 1.96·6-s + 3.41·8-s + 0.0271·9-s + 1.78·10-s + 0.637·11-s + 2.79·12-s − 0.277·13-s + 0.935·15-s + 3.85·16-s − 0.0446·17-s + 0.0527·18-s − 0.170·19-s + 2.54·20-s + 1.23·22-s + 0.820·23-s + 3.45·24-s − 0.147·25-s − 0.537·26-s − 0.985·27-s − 1.80·29-s + 1.81·30-s + 0.893·31-s + 4.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.35817953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.35817953\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 10.9T + 32T^{2} \) |
| 3 | \( 1 - 15.7T + 243T^{2} \) |
| 5 | \( 1 - 51.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 255.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 53.2T + 1.41e6T^{2} \) |
| 19 | \( 1 + 268.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.08e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.28e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.77e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.83e4T + 8.58e9T^{2} \) |
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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
−9.892937535080855165962898410838, −8.969139515817770411370697176119, −7.73854804157332967363429886096, −6.85432928728914000931700749284, −5.96176459192709643870173655013, −5.22516561330640488591407759066, −4.09738659278077744122156275283, −3.25587431845492114059232838820, −2.37362719704660493213253539591, −1.62102603336058343373203425217,
1.62102603336058343373203425217, 2.37362719704660493213253539591, 3.25587431845492114059232838820, 4.09738659278077744122156275283, 5.22516561330640488591407759066, 5.96176459192709643870173655013, 6.85432928728914000931700749284, 7.73854804157332967363429886096, 8.969139515817770411370697176119, 9.892937535080855165962898410838
