| L(s) = 1 | − 0.862·2-s − 3.07·3-s − 1.25·4-s + 2.65·6-s + 0.566·7-s + 2.80·8-s + 6.48·9-s − 1.91·11-s + 3.86·12-s − 0.194·13-s − 0.488·14-s + 0.0877·16-s + 5.29·17-s − 5.59·18-s − 1.74·21-s + 1.64·22-s + 3.37·23-s − 8.64·24-s + 0.167·26-s − 10.7·27-s − 0.711·28-s + 8.73·29-s − 5.65·31-s − 5.69·32-s + 5.88·33-s − 4.56·34-s − 8.13·36-s + ⋯ |
| L(s) = 1 | − 0.610·2-s − 1.77·3-s − 0.627·4-s + 1.08·6-s + 0.214·7-s + 0.993·8-s + 2.16·9-s − 0.576·11-s + 1.11·12-s − 0.0539·13-s − 0.130·14-s + 0.0219·16-s + 1.28·17-s − 1.31·18-s − 0.380·21-s + 0.351·22-s + 0.703·23-s − 1.76·24-s + 0.0329·26-s − 2.06·27-s − 0.134·28-s + 1.62·29-s − 1.01·31-s − 1.00·32-s + 1.02·33-s − 0.783·34-s − 1.35·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.862T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 - 0.566T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + 0.194T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 - 8.73T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 0.955T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 8.83T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 4.04T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 + 4.51T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.28456574353777095758656188681, −6.82229842429041668652964380992, −5.84187496488563231699223731425, −5.29992857853536408116500735609, −4.87837091753461205478655078647, −4.15368379720016700921155003243, −3.13539898880615646591987204513, −1.62099962742978483621219732820, −0.917805387583713856093890839137, 0,
0.917805387583713856093890839137, 1.62099962742978483621219732820, 3.13539898880615646591987204513, 4.15368379720016700921155003243, 4.87837091753461205478655078647, 5.29992857853536408116500735609, 5.84187496488563231699223731425, 6.82229842429041668652964380992, 7.28456574353777095758656188681
