| L(s) = 1 | + 2-s + 4-s − 1.17·5-s + 1.96·7-s + 8-s − 1.17·10-s − 0.465·11-s − 2.12·13-s + 1.96·14-s + 16-s + 0.590·17-s − 7.81·19-s − 1.17·20-s − 0.465·22-s + 4.36·23-s − 3.62·25-s − 2.12·26-s + 1.96·28-s − 1.44·29-s + 9.39·31-s + 32-s + 0.590·34-s − 2.30·35-s − 3.39·37-s − 7.81·38-s − 1.17·40-s − 12.4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.524·5-s + 0.741·7-s + 0.353·8-s − 0.370·10-s − 0.140·11-s − 0.589·13-s + 0.524·14-s + 0.250·16-s + 0.143·17-s − 1.79·19-s − 0.262·20-s − 0.0992·22-s + 0.911·23-s − 0.724·25-s − 0.416·26-s + 0.370·28-s − 0.267·29-s + 1.68·31-s + 0.176·32-s + 0.101·34-s − 0.388·35-s − 0.558·37-s − 1.26·38-s − 0.185·40-s − 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 + 0.465T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 - 0.590T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 9.39T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 - 0.711T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−7.51168167622621795050128057096, −6.67316508612091513529035646138, −6.16008263619035574339830363093, −5.12750704921019321318930432127, −4.69492884310265031491508823477, −4.05844183917842790265995115990, −3.17210789899354497059426356554, −2.33315891407980221618081312271, −1.47933780090759696100823868337, 0,
1.47933780090759696100823868337, 2.33315891407980221618081312271, 3.17210789899354497059426356554, 4.05844183917842790265995115990, 4.69492884310265031491508823477, 5.12750704921019321318930432127, 6.16008263619035574339830363093, 6.67316508612091513529035646138, 7.51168167622621795050128057096
