| L(s) = 1 | + 0.777·2-s − 1.39·4-s + 2.37·5-s − 2.50·7-s − 2.64·8-s + 1.84·10-s + 3.14·11-s + 1.33·13-s − 1.94·14-s + 0.736·16-s + 6.27·17-s + 8.06·19-s − 3.31·20-s + 2.44·22-s + 4.05·23-s + 0.647·25-s + 1.03·26-s + 3.48·28-s − 9.28·29-s + 2.83·31-s + 5.85·32-s + 4.87·34-s − 5.94·35-s + 5.53·37-s + 6.27·38-s − 6.27·40-s − 7.10·41-s + ⋯ |
| L(s) = 1 | + 0.549·2-s − 0.697·4-s + 1.06·5-s − 0.945·7-s − 0.933·8-s + 0.584·10-s + 0.947·11-s + 0.370·13-s − 0.519·14-s + 0.184·16-s + 1.52·17-s + 1.85·19-s − 0.741·20-s + 0.520·22-s + 0.845·23-s + 0.129·25-s + 0.203·26-s + 0.659·28-s − 1.72·29-s + 0.508·31-s + 1.03·32-s + 0.836·34-s − 1.00·35-s + 0.909·37-s + 1.01·38-s − 0.992·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958524500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958524500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 0.777T + 2T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 + 9.28T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 - 5.53T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 0.135T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 - 0.341T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 - 0.913T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−9.944575102831909031611897207087, −9.617617743035539646279669451666, −9.058750539178337873642419260258, −7.75617541647280971092780424542, −6.57900993088727946801195254603, −5.76317037820004068980446124346, −5.19334468670348608286477884622, −3.72501173676307956100822876708, −3.09116245082537454398632213836, −1.20821513091691800461851737607,
1.20821513091691800461851737607, 3.09116245082537454398632213836, 3.72501173676307956100822876708, 5.19334468670348608286477884622, 5.76317037820004068980446124346, 6.57900993088727946801195254603, 7.75617541647280971092780424542, 9.058750539178337873642419260258, 9.617617743035539646279669451666, 9.944575102831909031611897207087
