| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 2·13-s − 4·14-s + 16-s + 6·17-s − 4·19-s + 20-s + 25-s − 2·26-s − 4·28-s − 6·29-s + 32-s + 6·34-s − 4·35-s − 2·37-s − 4·38-s + 40-s + 6·41-s + 4·43-s + 9·49-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.26004129429744, −13.49293851254747, −13.16242935189219, −12.70748362340876, −12.36504201699858, −11.93065685833102, −11.20085525671168, −10.62949529556697, −10.18607552904297, −9.699666989429337, −9.327409537458839, −8.750123131285449, −7.903030374095728, −7.468604957931665, −6.913175776515394, −6.327756532282235, −5.962008698525998, −5.452579659058678, −4.890618624768287, −4.059729670501439, −3.637730625267883, −3.037610466538439, −2.510436476256027, −1.850936425640497, −0.9138977292020756, 0,
0.9138977292020756, 1.850936425640497, 2.510436476256027, 3.037610466538439, 3.637730625267883, 4.059729670501439, 4.890618624768287, 5.452579659058678, 5.962008698525998, 6.327756532282235, 6.913175776515394, 7.468604957931665, 7.903030374095728, 8.750123131285449, 9.327409537458839, 9.699666989429337, 10.18607552904297, 10.62949529556697, 11.20085525671168, 11.93065685833102, 12.36504201699858, 12.70748362340876, 13.16242935189219, 13.49293851254747, 14.26004129429744
