| L(s) = 1 | − 2.37·5-s + 2.20·7-s + 5.93·11-s + 4.37·13-s − 3.37·17-s − 3.72·19-s − 2.20·23-s + 0.627·25-s − 0.372·29-s + 9.66·31-s − 5.24·35-s + 4·37-s − 41-s + 5.93·43-s − 2.20·47-s − 2.11·49-s + 4·53-s − 14.0·55-s − 10.3·59-s + 15.1·61-s − 10.3·65-s − 10.3·67-s − 4.41·71-s + 4.62·73-s + 13.1·77-s − 9.66·79-s + 14.0·83-s + ⋯ |
| L(s) = 1 | − 1.06·5-s + 0.835·7-s + 1.78·11-s + 1.21·13-s − 0.817·17-s − 0.854·19-s − 0.460·23-s + 0.125·25-s − 0.0691·29-s + 1.73·31-s − 0.886·35-s + 0.657·37-s − 0.156·41-s + 0.905·43-s − 0.322·47-s − 0.302·49-s + 0.549·53-s − 1.89·55-s − 1.34·59-s + 1.93·61-s − 1.28·65-s − 1.26·67-s − 0.524·71-s + 0.541·73-s + 1.49·77-s − 1.08·79-s + 1.54·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850852211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850852211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.20T + 7T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 0.372T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 - 4.62T + 73T^{2} \) |
| 79 | \( 1 + 9.66T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−8.606226379168249952026368775663, −8.333283757973059859915617246706, −7.43073950856979624177135422654, −6.49873116795932681862682809972, −6.04322221251578386800795532402, −4.51074923520144160003334820886, −4.24540646108783905491131066318, −3.41472776977553945315954108179, −1.96691469883615189453250340838, −0.906323897796769393764080167284,
0.906323897796769393764080167284, 1.96691469883615189453250340838, 3.41472776977553945315954108179, 4.24540646108783905491131066318, 4.51074923520144160003334820886, 6.04322221251578386800795532402, 6.49873116795932681862682809972, 7.43073950856979624177135422654, 8.333283757973059859915617246706, 8.606226379168249952026368775663
