| L(s) = 1 | + 0.523·2-s − 1.72·4-s − 2.20·5-s + 7-s − 1.95·8-s − 1.15·10-s + 5.20·11-s + 3.15·13-s + 0.523·14-s + 2.42·16-s − 3.24·17-s + 7.45·19-s + 3.79·20-s + 2.72·22-s + 4.40·23-s − 0.153·25-s + 1.65·26-s − 1.72·28-s − 1.15·29-s + 2·31-s + 5.17·32-s − 1.70·34-s − 2.20·35-s + 5·37-s + 3.90·38-s + 4.29·40-s − 11.4·41-s + ⋯ |
| L(s) = 1 | + 0.370·2-s − 0.862·4-s − 0.984·5-s + 0.377·7-s − 0.690·8-s − 0.364·10-s + 1.56·11-s + 0.874·13-s + 0.140·14-s + 0.607·16-s − 0.788·17-s + 1.70·19-s + 0.849·20-s + 0.581·22-s + 0.918·23-s − 0.0307·25-s + 0.324·26-s − 0.326·28-s − 0.214·29-s + 0.359·31-s + 0.915·32-s − 0.291·34-s − 0.372·35-s + 0.821·37-s + 0.633·38-s + 0.679·40-s − 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.357589311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357589311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.523T + 2T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 + 0.249T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 - 9.60T + 71T^{2} \) |
| 73 | \( 1 - 0.846T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 9.24T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−11.05933668452361786763920730696, −9.625463589188784421716292238724, −8.972449276476250496814274501557, −8.211581357327969110328321628474, −7.18211272805310602846537948230, −6.09970439573613097322423646176, −4.92527118924494816110086204898, −4.04335696035270554747747997918, −3.35419590831355860192797384487, −1.06737710998401794297751551002,
1.06737710998401794297751551002, 3.35419590831355860192797384487, 4.04335696035270554747747997918, 4.92527118924494816110086204898, 6.09970439573613097322423646176, 7.18211272805310602846537948230, 8.211581357327969110328321628474, 8.972449276476250496814274501557, 9.625463589188784421716292238724, 11.05933668452361786763920730696
