| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 4·11-s + 13-s + 15-s − 2·17-s − 4·19-s + 4·21-s − 6·23-s + 25-s − 27-s − 29-s − 8·31-s − 4·33-s + 4·35-s − 10·37-s − 39-s − 6·41-s + 8·43-s − 45-s − 10·47-s + 9·49-s + 2·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 1.64·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 1.45·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.78249242813891, −12.59710083387598, −12.01399151243734, −11.73668866840911, −11.15932413897047, −10.77375525630657, −10.16233553354944, −9.969107920572720, −9.233927963057145, −8.994170094420843, −8.562056240870578, −7.892692720069573, −7.272317150005755, −6.880340264850848, −6.452995001611472, −6.139415066934815, −5.697950669644783, −4.983450593849749, −4.402053462935719, −3.866606856369570, −3.551713069511255, −3.156689237491867, −2.092941981675342, −1.841163175845666, −0.9378495034913775, 0, 0,
0.9378495034913775, 1.841163175845666, 2.092941981675342, 3.156689237491867, 3.551713069511255, 3.866606856369570, 4.402053462935719, 4.983450593849749, 5.697950669644783, 6.139415066934815, 6.452995001611472, 6.880340264850848, 7.272317150005755, 7.892692720069573, 8.562056240870578, 8.994170094420843, 9.233927963057145, 9.969107920572720, 10.16233553354944, 10.77375525630657, 11.15932413897047, 11.73668866840911, 12.01399151243734, 12.59710083387598, 12.78249242813891
