| L(s) = 1 | + 5-s − 0.648·7-s − 3.35·11-s + 4.17·13-s − 4.82·17-s + 6.82·19-s − 5.52·23-s + 25-s + 29-s − 2.82·31-s − 0.648·35-s − 10.2·37-s − 8.17·41-s − 5.69·43-s + 2.64·47-s − 6.58·49-s + 2.87·53-s − 3.35·55-s + 13.2·59-s − 1.12·61-s + 4.17·65-s + 1.52·67-s + 8.87·71-s + 9.69·73-s + 2.17·77-s + 8.99·79-s − 1.94·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.244·7-s − 1.01·11-s + 1.15·13-s − 1.16·17-s + 1.56·19-s − 1.15·23-s + 0.200·25-s + 0.185·29-s − 0.506·31-s − 0.109·35-s − 1.68·37-s − 1.27·41-s − 0.868·43-s + 0.386·47-s − 0.940·49-s + 0.395·53-s − 0.451·55-s + 1.72·59-s − 0.143·61-s + 0.517·65-s + 0.186·67-s + 1.05·71-s + 1.13·73-s + 0.247·77-s + 1.01·79-s − 0.213·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 0.648T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.041227389211241973804789008303, −6.92687192791166052311481522380, −6.56033229582637809809321863363, −5.42716492047027328077659318231, −5.26283226594296558626517902941, −3.99316201191640441343857693944, −3.31387031175211363420609494670, −2.34865965233774095571105588383, −1.44087116678613878655112741439, 0,
1.44087116678613878655112741439, 2.34865965233774095571105588383, 3.31387031175211363420609494670, 3.99316201191640441343857693944, 5.26283226594296558626517902941, 5.42716492047027328077659318231, 6.56033229582637809809321863363, 6.92687192791166052311481522380, 8.041227389211241973804789008303
