| L(s) = 1 | − 2.74·5-s + 0.601·7-s − 3·11-s − 3.29·13-s + 2.14·17-s + 5.89·19-s − 1.60·23-s + 2.54·25-s + 5.14·29-s + 3.14·31-s − 1.65·35-s + 4.43·37-s − 11.1·41-s + 3·43-s − 1.65·47-s − 6.63·49-s + 6.74·53-s + 8.23·55-s + 59-s + 3.43·61-s + 9.03·65-s − 1.94·67-s − 4.70·71-s + 10.9·73-s − 1.80·77-s − 0.202·79-s − 3.39·83-s + ⋯ |
| L(s) = 1 | − 1.22·5-s + 0.227·7-s − 0.904·11-s − 0.912·13-s + 0.520·17-s + 1.35·19-s − 0.333·23-s + 0.508·25-s + 0.955·29-s + 0.564·31-s − 0.279·35-s + 0.729·37-s − 1.74·41-s + 0.457·43-s − 0.240·47-s − 0.948·49-s + 0.926·53-s + 1.11·55-s + 0.130·59-s + 0.439·61-s + 1.12·65-s − 0.238·67-s − 0.558·71-s + 1.27·73-s − 0.205·77-s − 0.0228·79-s − 0.373·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 - 0.601T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 61 | \( 1 - 3.43T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 0.202T + 79T^{2} \) |
| 83 | \( 1 + 3.39T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.52550937035989915513086220676, −7.01863483534592147927553451858, −6.04669213459901437413180355927, −5.07831242649041624940796465314, −4.81438877626642589776093087581, −3.81054752035671358869296450635, −3.13582522830935349710939599336, −2.37954470101896408434807881002, −1.05646554029782708097205442427, 0,
1.05646554029782708097205442427, 2.37954470101896408434807881002, 3.13582522830935349710939599336, 3.81054752035671358869296450635, 4.81438877626642589776093087581, 5.07831242649041624940796465314, 6.04669213459901437413180355927, 7.01863483534592147927553451858, 7.52550937035989915513086220676
