| L(s) = 1 | + 1.28·2-s + 18.9·3-s − 30.3·4-s − 25·5-s + 24.4·6-s − 83.2·7-s − 80.2·8-s + 117.·9-s − 32.1·10-s − 121·11-s − 576.·12-s + 600.·13-s − 107.·14-s − 474.·15-s + 867.·16-s + 2.06e3·17-s + 151.·18-s − 361·19-s + 758.·20-s − 1.58e3·21-s − 155.·22-s − 724.·23-s − 1.52e3·24-s + 625·25-s + 772.·26-s − 2.38e3·27-s + 2.52e3·28-s + ⋯ |
| L(s) = 1 | + 0.227·2-s + 1.21·3-s − 0.948·4-s − 0.447·5-s + 0.277·6-s − 0.641·7-s − 0.443·8-s + 0.483·9-s − 0.101·10-s − 0.301·11-s − 1.15·12-s + 0.985·13-s − 0.146·14-s − 0.544·15-s + 0.847·16-s + 1.73·17-s + 0.110·18-s − 0.229·19-s + 0.424·20-s − 0.781·21-s − 0.0686·22-s − 0.285·23-s − 0.540·24-s + 0.200·25-s + 0.224·26-s − 0.628·27-s + 0.608·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 1.28T + 32T^{2} \) |
| 3 | \( 1 - 18.9T + 243T^{2} \) |
| 7 | \( 1 + 83.2T + 1.68e4T^{2} \) |
| 13 | \( 1 - 600.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.06e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 724.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 267.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.87e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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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
−8.774417639301812926035543734374, −8.057774016855896646428110941124, −7.47097911624748504894067925936, −6.07010961448751715638663301195, −5.30369162069493620372495443335, −3.90595844770343797248843418957, −3.59738861611742703358133085580, −2.70582565773225299364786305090, −1.20680759625476926887032175343, 0,
1.20680759625476926887032175343, 2.70582565773225299364786305090, 3.59738861611742703358133085580, 3.90595844770343797248843418957, 5.30369162069493620372495443335, 6.07010961448751715638663301195, 7.47097911624748504894067925936, 8.057774016855896646428110941124, 8.774417639301812926035543734374
