| L(s) = 1 | + 4.20·2-s − 9·3-s − 14.3·4-s − 81.8·5-s − 37.8·6-s + 101.·7-s − 194.·8-s + 81·9-s − 344.·10-s + 52.0·11-s + 129.·12-s + 129.·13-s + 426.·14-s + 737.·15-s − 359.·16-s + 1.25e3·17-s + 340.·18-s + 2.12e3·19-s + 1.17e3·20-s − 914.·21-s + 218.·22-s + 2.25e3·23-s + 1.75e3·24-s + 3.58e3·25-s + 544.·26-s − 729·27-s − 1.45e3·28-s + ⋯ |
| L(s) = 1 | + 0.742·2-s − 0.577·3-s − 0.448·4-s − 1.46·5-s − 0.428·6-s + 0.783·7-s − 1.07·8-s + 0.333·9-s − 1.08·10-s + 0.129·11-s + 0.258·12-s + 0.212·13-s + 0.581·14-s + 0.845·15-s − 0.351·16-s + 1.05·17-s + 0.247·18-s + 1.35·19-s + 0.656·20-s − 0.452·21-s + 0.0963·22-s + 0.889·23-s + 0.621·24-s + 1.14·25-s + 0.157·26-s − 0.192·27-s − 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 4.20T + 32T^{2} \) |
| 5 | \( 1 + 81.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 52.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 129.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 727.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.56e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.93e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.30e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.68e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.64e4T + 8.58e9T^{2} \) |
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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
−9.809429658444649148969908305018, −8.755445406308491357980537822938, −7.82090191841817423729111218559, −7.08309266106877553075967222774, −5.55465720860498530115620458459, −5.02538591405188060976615417248, −3.94800166722276360045607081115, −3.30630982391857875079715837822, −1.16301809417008211318311700128, 0,
1.16301809417008211318311700128, 3.30630982391857875079715837822, 3.94800166722276360045607081115, 5.02538591405188060976615417248, 5.55465720860498530115620458459, 7.08309266106877553075967222774, 7.82090191841817423729111218559, 8.755445406308491357980537822938, 9.809429658444649148969908305018
