| L(s) = 1 | − 2.44·2-s + 4.00·4-s − 0.217·5-s − 1.53·7-s − 4.90·8-s + 0.534·10-s − 1.30·11-s − 3.76·13-s + 3.75·14-s + 4.01·16-s + 5.99·17-s + 5.43·19-s − 0.872·20-s + 3.20·22-s + 6.21·23-s − 4.95·25-s + 9.22·26-s − 6.13·28-s − 7.36·31-s − 0.0182·32-s − 14.6·34-s + 0.334·35-s − 5.08·37-s − 13.3·38-s + 1.06·40-s + 8.96·41-s − 4.20·43-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.00·4-s − 0.0974·5-s − 0.579·7-s − 1.73·8-s + 0.168·10-s − 0.394·11-s − 1.04·13-s + 1.00·14-s + 1.00·16-s + 1.45·17-s + 1.24·19-s − 0.195·20-s + 0.683·22-s + 1.29·23-s − 0.990·25-s + 1.80·26-s − 1.16·28-s − 1.32·31-s − 0.00322·32-s − 2.51·34-s + 0.0565·35-s − 0.835·37-s − 2.15·38-s + 0.169·40-s + 1.40·41-s − 0.641·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 + 0.217T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 0.398T + 47T^{2} \) |
| 53 | \( 1 + 8.62T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 3.60T + 79T^{2} \) |
| 83 | \( 1 + 2.91T + 83T^{2} \) |
| 89 | \( 1 + 0.433T + 89T^{2} \) |
| 97 | \( 1 - 6.04T + 97T^{2} \) |
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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
−7.61770728385987645942544116524, −7.26588974985635003467753209645, −6.46625262331733143728679751224, −5.56265165008372724372676040783, −4.94182519927531263997836732207, −3.49676223144492492247469890688, −2.94718961065902513218671570357, −1.96363785027452383050699679512, −1.00063205388508171041685843812, 0,
1.00063205388508171041685843812, 1.96363785027452383050699679512, 2.94718961065902513218671570357, 3.49676223144492492247469890688, 4.94182519927531263997836732207, 5.56265165008372724372676040783, 6.46625262331733143728679751224, 7.26588974985635003467753209645, 7.61770728385987645942544116524
