| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 4·11-s + 15-s − 17-s + 4·21-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s + 4·35-s − 10·37-s − 2·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s + 51-s + 6·53-s + 4·55-s − 4·59-s − 6·61-s − 4·63-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s − 0.242·17-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 0.768·61-s − 0.503·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.96964248988443, −12.53094584781914, −12.13796955071718, −11.83618837902448, −11.03762177888958, −10.74946045039161, −10.33771357061874, −9.980711497979775, −9.373395148374618, −9.026765932424709, −8.401472913173044, −7.952845155160709, −7.381943714581502, −6.794018646145445, −6.766417554219218, −5.915143562899391, −5.659931507208247, −5.016447446969416, −4.634477670952543, −3.902720152746224, −3.409909588857762, −3.032146437964930, −2.405056619108503, −1.748172788237585, −0.8657278689994047, 0, 0,
0.8657278689994047, 1.748172788237585, 2.405056619108503, 3.032146437964930, 3.409909588857762, 3.902720152746224, 4.634477670952543, 5.016447446969416, 5.659931507208247, 5.915143562899391, 6.766417554219218, 6.794018646145445, 7.381943714581502, 7.952845155160709, 8.401472913173044, 9.026765932424709, 9.373395148374618, 9.980711497979775, 10.33771357061874, 10.74946045039161, 11.03762177888958, 11.83618837902448, 12.13796955071718, 12.53094584781914, 12.96964248988443
