| L(s) = 1 | − 2.56·5-s + 2.56·7-s − 1.43·11-s − 5.12·13-s + 5.68·17-s + 19-s − 0.876·23-s + 1.56·25-s − 8.24·29-s − 2·31-s − 6.56·35-s − 8·37-s − 3.12·41-s + 2.56·43-s − 5.68·47-s − 0.438·49-s + 12.2·53-s + 3.68·55-s − 12·59-s − 5.68·61-s + 13.1·65-s − 10.2·67-s + 11.9·73-s − 3.68·77-s − 13.3·79-s − 4·83-s − 14.5·85-s + ⋯ |
| L(s) = 1 | − 1.14·5-s + 0.968·7-s − 0.433·11-s − 1.42·13-s + 1.37·17-s + 0.229·19-s − 0.182·23-s + 0.312·25-s − 1.53·29-s − 0.359·31-s − 1.10·35-s − 1.31·37-s − 0.487·41-s + 0.390·43-s − 0.829·47-s − 0.0626·49-s + 1.68·53-s + 0.496·55-s − 1.56·59-s − 0.727·61-s + 1.62·65-s − 1.25·67-s + 1.39·73-s − 0.419·77-s − 1.50·79-s − 0.439·83-s − 1.57·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 23 | \( 1 + 0.876T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 5.68T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−9.149236607541663322133941012394, −8.084711086656233871244845227047, −7.66190770990152056405132713219, −7.12119993922417367778747557959, −5.58047950761476600184648104997, −4.98739524312174120048643645785, −4.02883090227823545932008734574, −3.06940715612023451468291156546, −1.72180829274827917643095105983, 0,
1.72180829274827917643095105983, 3.06940715612023451468291156546, 4.02883090227823545932008734574, 4.98739524312174120048643645785, 5.58047950761476600184648104997, 7.12119993922417367778747557959, 7.66190770990152056405132713219, 8.084711086656233871244845227047, 9.149236607541663322133941012394
