| L(s) = 1 | + 2.64·2-s + 4.98·4-s + 0.579·5-s − 5.09·7-s + 7.89·8-s + 1.53·10-s − 11-s + 0.741·13-s − 13.4·14-s + 10.8·16-s − 6.81·17-s + 3.91·19-s + 2.89·20-s − 2.64·22-s − 2.39·23-s − 4.66·25-s + 1.95·26-s − 25.3·28-s − 8.20·29-s + 3.98·31-s + 13.0·32-s − 18.0·34-s − 2.95·35-s − 0.211·37-s + 10.3·38-s + 4.57·40-s − 1.22·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.49·4-s + 0.259·5-s − 1.92·7-s + 2.79·8-s + 0.484·10-s − 0.301·11-s + 0.205·13-s − 3.59·14-s + 2.72·16-s − 1.65·17-s + 0.898·19-s + 0.646·20-s − 0.563·22-s − 0.498·23-s − 0.932·25-s + 0.384·26-s − 4.79·28-s − 1.52·29-s + 0.715·31-s + 2.30·32-s − 3.09·34-s − 0.498·35-s − 0.0348·37-s + 1.67·38-s + 0.723·40-s − 0.191·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 - 0.579T + 5T^{2} \) |
| 7 | \( 1 + 5.09T + 7T^{2} \) |
| 13 | \( 1 - 0.741T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 0.211T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 9.68T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 2.29T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 + 0.0348T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 1.77T + 71T^{2} \) |
| 73 | \( 1 + 0.244T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.01664982735336311035400174668, −6.52112149852699356140147666153, −6.07351561455047335218966207244, −5.42816983971346027982950429459, −4.63977548248183571824494820684, −3.76138747506933600894192283150, −3.37539113131122374701394614706, −2.58114379259172054084584318643, −1.84218860253485143201241159920, 0,
1.84218860253485143201241159920, 2.58114379259172054084584318643, 3.37539113131122374701394614706, 3.76138747506933600894192283150, 4.63977548248183571824494820684, 5.42816983971346027982950429459, 6.07351561455047335218966207244, 6.52112149852699356140147666153, 7.01664982735336311035400174668
