| L(s) = 1 | + 1.77·2-s + 9·3-s − 28.8·4-s − 4.92·5-s + 16.0·6-s − 49·7-s − 108.·8-s + 81·9-s − 8.76·10-s + 738.·11-s − 259.·12-s − 63.0·13-s − 87.1·14-s − 44.3·15-s + 730.·16-s − 776.·17-s + 144.·18-s − 2.58e3·19-s + 142.·20-s − 441·21-s + 1.31e3·22-s + 529·23-s − 974.·24-s − 3.10e3·25-s − 112.·26-s + 729·27-s + 1.41e3·28-s + ⋯ |
| L(s) = 1 | + 0.314·2-s + 0.577·3-s − 0.901·4-s − 0.0881·5-s + 0.181·6-s − 0.377·7-s − 0.597·8-s + 0.333·9-s − 0.0277·10-s + 1.84·11-s − 0.520·12-s − 0.103·13-s − 0.118·14-s − 0.0508·15-s + 0.713·16-s − 0.651·17-s + 0.104·18-s − 1.64·19-s + 0.0794·20-s − 0.218·21-s + 0.578·22-s + 0.208·23-s − 0.345·24-s − 0.992·25-s − 0.0325·26-s + 0.192·27-s + 0.340·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 7 | \( 1 + 49T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 1.77T + 32T^{2} \) |
| 5 | \( 1 + 4.92T + 3.12e3T^{2} \) |
| 11 | \( 1 - 738.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 63.0T + 3.71e5T^{2} \) |
| 17 | \( 1 + 776.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.58e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.17e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.43e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.02e4T + 8.58e9T^{2} \) |
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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
−9.580165174088846560995468414275, −8.801908155932799976757838294329, −8.282074387779111935864712382628, −6.76223841654517476533550333230, −6.15032942987494478202643345488, −4.51802344973832360218309153629, −4.08398240579092745244590025628, −2.93799953458970624436610269433, −1.43204055630892588742907962460, 0,
1.43204055630892588742907962460, 2.93799953458970624436610269433, 4.08398240579092745244590025628, 4.51802344973832360218309153629, 6.15032942987494478202643345488, 6.76223841654517476533550333230, 8.282074387779111935864712382628, 8.801908155932799976757838294329, 9.580165174088846560995468414275
