| L(s) = 1 | + 1.12·2-s − 0.741·4-s + 1.73·5-s − 0.583·7-s − 3.07·8-s + 1.94·10-s − 3.27·11-s + 5.76·13-s − 0.654·14-s − 1.96·16-s − 6.22·17-s − 0.717·19-s − 1.28·20-s − 3.67·22-s + 3.91·23-s − 1.98·25-s + 6.46·26-s + 0.432·28-s − 1.99·29-s + 3.94·32-s − 6.98·34-s − 1.01·35-s + 7.34·37-s − 0.804·38-s − 5.34·40-s + 3.13·41-s − 6.22·43-s + ⋯ |
| L(s) = 1 | + 0.793·2-s − 0.370·4-s + 0.776·5-s − 0.220·7-s − 1.08·8-s + 0.616·10-s − 0.986·11-s + 1.59·13-s − 0.174·14-s − 0.491·16-s − 1.51·17-s − 0.164·19-s − 0.288·20-s − 0.782·22-s + 0.816·23-s − 0.396·25-s + 1.26·26-s + 0.0817·28-s − 0.369·29-s + 0.697·32-s − 1.19·34-s − 0.171·35-s + 1.20·37-s − 0.130·38-s − 0.844·40-s + 0.489·41-s − 0.949·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.433002079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433002079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 0.583T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 + 0.717T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 + 1.99T + 29T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 + 6.22T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 - 0.466T + 53T^{2} \) |
| 59 | \( 1 - 0.707T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 4.03T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 8.65T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−7.890767577076209373628848745086, −6.63621713336365918996576386002, −6.36492365008425847686976481342, −5.57258953006478551824106992099, −5.09022733421359382524959309314, −4.24303876359450929227965966259, −3.61403298894724995241461408332, −2.74460766773673173943155119345, −1.99433241727767624407255999166, −0.65942192681524416116724945132,
0.65942192681524416116724945132, 1.99433241727767624407255999166, 2.74460766773673173943155119345, 3.61403298894724995241461408332, 4.24303876359450929227965966259, 5.09022733421359382524959309314, 5.57258953006478551824106992099, 6.36492365008425847686976481342, 6.63621713336365918996576386002, 7.890767577076209373628848745086
