| L(s) = 1 | − 4·7-s − 6·11-s − 13-s − 6·17-s + 4·19-s − 3·23-s − 3·29-s − 4·31-s − 2·37-s − 6·41-s + 7·43-s + 9·49-s − 9·53-s − 6·59-s + 61-s − 14·67-s + 6·71-s − 4·73-s + 24·77-s + 11·79-s − 6·83-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.625·23-s − 0.557·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.06·43-s + 9/7·49-s − 1.23·53-s − 0.781·59-s + 0.128·61-s − 1.71·67-s + 0.712·71-s − 0.468·73-s + 2.73·77-s + 1.23·79-s − 0.658·83-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.21254282559560, −13.01188942304998, −12.48782536954160, −12.13427252805638, −11.42364819381411, −10.86937076655324, −10.56435498241791, −10.05944086320857, −9.492852764519771, −9.296583795240266, −8.639679063274785, −8.038157621581098, −7.562663546155064, −7.102312671567247, −6.654897622469223, −5.977747441792864, −5.672383334830426, −5.035132576915433, −4.540050711940085, −3.846047673895222, −3.216900804143552, −2.850478773829382, −2.276686581563119, −1.655720409998415, −0.4443980468744329, 0,
0.4443980468744329, 1.655720409998415, 2.276686581563119, 2.850478773829382, 3.216900804143552, 3.846047673895222, 4.540050711940085, 5.035132576915433, 5.672383334830426, 5.977747441792864, 6.654897622469223, 7.102312671567247, 7.562663546155064, 8.038157621581098, 8.639679063274785, 9.296583795240266, 9.492852764519771, 10.05944086320857, 10.56435498241791, 10.86937076655324, 11.42364819381411, 12.13427252805638, 12.48782536954160, 13.01188942304998, 13.21254282559560
