| L(s) = 1 | + 0.414·2-s − 1.82·4-s − 5-s − 1.41·7-s − 1.58·8-s − 0.414·10-s − 4·11-s + 4.82·13-s − 0.585·14-s + 3·16-s − 1.17·17-s + 6.24·19-s + 1.82·20-s − 1.65·22-s − 0.242·23-s + 25-s + 1.99·26-s + 2.58·28-s + 3.17·29-s + 0.585·31-s + 4.41·32-s − 0.485·34-s + 1.41·35-s + 7.65·37-s + 2.58·38-s + 1.58·40-s − 0.828·41-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.534·7-s − 0.560·8-s − 0.130·10-s − 1.20·11-s + 1.33·13-s − 0.156·14-s + 0.750·16-s − 0.284·17-s + 1.43·19-s + 0.408·20-s − 0.353·22-s − 0.0505·23-s + 0.200·25-s + 0.392·26-s + 0.488·28-s + 0.588·29-s + 0.105·31-s + 0.780·32-s − 0.0832·34-s + 0.239·35-s + 1.25·37-s + 0.419·38-s + 0.250·40-s − 0.129·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 0.242T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 0.585T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 5.17T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−8.149077403002419476408787458920, −7.51461815968400769368963670210, −6.45586849826137912966710300832, −5.76735200270194737929520697983, −5.02465593409263011522478162976, −4.29758573902045794306812230495, −3.37477798140652192905358209618, −2.89881801746732864196894754881, −1.19341505803828322845270986129, 0,
1.19341505803828322845270986129, 2.89881801746732864196894754881, 3.37477798140652192905358209618, 4.29758573902045794306812230495, 5.02465593409263011522478162976, 5.76735200270194737929520697983, 6.45586849826137912966710300832, 7.51461815968400769368963670210, 8.149077403002419476408787458920
