| L(s) = 1 | − 2.63·2-s + 3-s + 4.95·4-s + 2.22·5-s − 2.63·6-s + 2.31·7-s − 7.77·8-s + 9-s − 5.85·10-s − 1.95·11-s + 4.95·12-s + 1.32·13-s − 6.10·14-s + 2.22·15-s + 10.6·16-s − 2.63·18-s + 5.95·19-s + 11.0·20-s + 2.31·21-s + 5.14·22-s + 4.77·23-s − 7.77·24-s − 0.0618·25-s − 3.48·26-s + 27-s + 11.4·28-s − 0.172·29-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.47·4-s + 0.993·5-s − 1.07·6-s + 0.874·7-s − 2.75·8-s + 0.333·9-s − 1.85·10-s − 0.588·11-s + 1.42·12-s + 0.366·13-s − 1.63·14-s + 0.573·15-s + 2.65·16-s − 0.621·18-s + 1.36·19-s + 2.45·20-s + 0.505·21-s + 1.09·22-s + 0.996·23-s − 1.58·24-s − 0.0123·25-s − 0.683·26-s + 0.192·27-s + 2.16·28-s − 0.0320·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131519006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131519006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 0.172T + 29T^{2} \) |
| 31 | \( 1 + 0.444T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 0.727T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 0.698T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−9.953335904917807539819830722058, −9.239207315223529626818258746654, −8.669021678197057230538516659576, −7.77172358372860237187741659789, −7.24664193522977241612057758121, −6.10587112419350718253679559959, −5.10177025621254452958284314528, −3.13460755696983102245469158780, −2.09442863889157048303013266601, −1.21342119727785136109879772057,
1.21342119727785136109879772057, 2.09442863889157048303013266601, 3.13460755696983102245469158780, 5.10177025621254452958284314528, 6.10587112419350718253679559959, 7.24664193522977241612057758121, 7.77172358372860237187741659789, 8.669021678197057230538516659576, 9.239207315223529626818258746654, 9.953335904917807539819830722058
