| L(s) = 1 | + 3·3-s + 5·5-s − 4·7-s + 9·9-s + 6·11-s − 44·13-s + 15·15-s − 84·17-s + 8·19-s − 12·21-s + 48·23-s + 25·25-s + 27·27-s + 90·29-s + 166·31-s + 18·33-s − 20·35-s + 156·37-s − 132·39-s + 166·41-s − 460·43-s + 45·45-s − 448·47-s − 327·49-s − 252·51-s − 470·53-s + 30·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.215·7-s + 1/3·9-s + 0.164·11-s − 0.938·13-s + 0.258·15-s − 1.19·17-s + 0.0965·19-s − 0.124·21-s + 0.435·23-s + 1/5·25-s + 0.192·27-s + 0.576·29-s + 0.961·31-s + 0.0949·33-s − 0.0965·35-s + 0.693·37-s − 0.541·39-s + 0.632·41-s − 1.63·43-s + 0.149·45-s − 1.39·47-s − 0.953·49-s − 0.691·51-s − 1.21·53-s + 0.0735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 166 T + p^{3} T^{2} \) |
| 37 | \( 1 - 156 T + p^{3} T^{2} \) |
| 41 | \( 1 - 166 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 + 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 470 T + p^{3} T^{2} \) |
| 59 | \( 1 - 26 T + p^{3} T^{2} \) |
| 61 | \( 1 + 206 T + p^{3} T^{2} \) |
| 67 | \( 1 + 548 T + p^{3} T^{2} \) |
| 71 | \( 1 + 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 30 T + p^{3} T^{2} \) |
| 79 | \( 1 - 750 T + p^{3} T^{2} \) |
| 83 | \( 1 + 4 T + p^{3} T^{2} \) |
| 89 | \( 1 + 186 T + p^{3} T^{2} \) |
| 97 | \( 1 - 530 T + p^{3} T^{2} \) |
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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
−8.476615900912315716821210782118, −7.76944096031601124412076969451, −6.76695588459536896189528281869, −6.31339805471618172260482437353, −5.00626379445994919911324074001, −4.45284293965903330759314389145, −3.19262382244758030766663413678, −2.47536753956834499661688244967, −1.45093381998494233024173577530, 0,
1.45093381998494233024173577530, 2.47536753956834499661688244967, 3.19262382244758030766663413678, 4.45284293965903330759314389145, 5.00626379445994919911324074001, 6.31339805471618172260482437353, 6.76695588459536896189528281869, 7.76944096031601124412076969451, 8.476615900912315716821210782118
