| L(s) = 1 | − 1.85·2-s + 1.43·4-s − 5-s − 0.0119·7-s + 1.05·8-s + 1.85·10-s + 3.66·11-s + 0.322·13-s + 0.0221·14-s − 4.81·16-s − 3.56·17-s − 3.88·19-s − 1.43·20-s − 6.79·22-s + 3.76·23-s + 25-s − 0.597·26-s − 0.0170·28-s + 10.2·29-s − 10.3·31-s + 6.80·32-s + 6.60·34-s + 0.0119·35-s + 2.45·37-s + 7.18·38-s − 1.05·40-s − 6.31·41-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.715·4-s − 0.447·5-s − 0.00450·7-s + 0.372·8-s + 0.585·10-s + 1.10·11-s + 0.0895·13-s + 0.00590·14-s − 1.20·16-s − 0.864·17-s − 0.890·19-s − 0.319·20-s − 1.44·22-s + 0.785·23-s + 0.200·25-s − 0.117·26-s − 0.00322·28-s + 1.90·29-s − 1.86·31-s + 1.20·32-s + 1.13·34-s + 0.00201·35-s + 0.403·37-s + 1.16·38-s − 0.166·40-s − 0.986·41-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 | \( 1 + 1.85T + 2T^{2} \) |
| 7 | \( 1 + 0.0119T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 0.322T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.04T + 43T^{2} \) |
| 47 | \( 1 + 9.55T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 + 6.03T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 - 3.56T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 + 0.742T + 83T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.451745745430590355116419270523, −7.44953213196436308060536099526, −6.79712910456861448127769631746, −6.29864144899535732547558579136, −4.90498729826696848589408969273, −4.30891536994358317110331520398, −3.33930849831895019821394093174, −2.09988558949715789938253478662, −1.18753456953819428695905297581, 0,
1.18753456953819428695905297581, 2.09988558949715789938253478662, 3.33930849831895019821394093174, 4.30891536994358317110331520398, 4.90498729826696848589408969273, 6.29864144899535732547558579136, 6.79712910456861448127769631746, 7.44953213196436308060536099526, 8.451745745430590355116419270523
