| L(s) = 1 | + 7.62·2-s − 7.42·3-s + 26.1·4-s − 25·5-s − 56.6·6-s − 46.3·7-s − 44.2·8-s − 187.·9-s − 190.·10-s − 121·11-s − 194.·12-s + 680.·13-s − 353.·14-s + 185.·15-s − 1.17e3·16-s − 224.·17-s − 1.43e3·18-s + 361·19-s − 654.·20-s + 344.·21-s − 923.·22-s − 2.28e3·23-s + 328.·24-s + 625·25-s + 5.18e3·26-s + 3.20e3·27-s − 1.21e3·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.476·3-s + 0.818·4-s − 0.447·5-s − 0.642·6-s − 0.357·7-s − 0.244·8-s − 0.772·9-s − 0.603·10-s − 0.301·11-s − 0.390·12-s + 1.11·13-s − 0.482·14-s + 0.213·15-s − 1.14·16-s − 0.188·17-s − 1.04·18-s + 0.229·19-s − 0.366·20-s + 0.170·21-s − 0.406·22-s − 0.901·23-s + 0.116·24-s + 0.200·25-s + 1.50·26-s + 0.844·27-s − 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.815001941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815001941\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 7.62T + 32T^{2} \) |
| 3 | \( 1 + 7.42T + 243T^{2} \) |
| 7 | \( 1 + 46.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 680.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 224.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.52e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.11e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.33e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.55e4T + 8.58e9T^{2} \) |
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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
−9.052718725248782558256218158342, −8.387701145744393250309890334883, −7.20764492169585351438316070378, −6.27100065248664466179592792047, −5.68735330005422789993153914908, −4.95974653771510056313177435159, −3.81208311536548219650289302638, −3.35521827074826745217980303218, −2.13259867934357492546089817857, −0.46023653298570170434455396088,
0.46023653298570170434455396088, 2.13259867934357492546089817857, 3.35521827074826745217980303218, 3.81208311536548219650289302638, 4.95974653771510056313177435159, 5.68735330005422789993153914908, 6.27100065248664466179592792047, 7.20764492169585351438316070378, 8.387701145744393250309890334883, 9.052718725248782558256218158342
