| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 7-s − 8-s + 9-s − 2·10-s + 3·11-s + 12-s + 6·13-s + 14-s + 2·15-s + 16-s − 18-s + 19-s + 2·20-s − 21-s − 3·22-s − 23-s − 24-s − 25-s − 6·26-s + 27-s − 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.447·20-s − 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.08986900908663, −12.57557405790202, −12.05084540620439, −11.48410623286924, −10.93452381039203, −10.73166705753032, −10.13130116663805, −9.518392708760017, −9.232143552801635, −9.021819360267459, −8.536115003591576, −7.787648961376502, −7.566269476348279, −6.870027021650821, −6.352377478108392, −6.056336424387775, −5.588924049691986, −4.968487737065649, −3.961372389356609, −3.734960401772952, −3.317862815377286, −2.458594548716402, −1.964334076355952, −1.467368319008718, −0.9935686474597030, 0,
0.9935686474597030, 1.467368319008718, 1.964334076355952, 2.458594548716402, 3.317862815377286, 3.734960401772952, 3.961372389356609, 4.968487737065649, 5.588924049691986, 6.056336424387775, 6.352377478108392, 6.870027021650821, 7.566269476348279, 7.787648961376502, 8.536115003591576, 9.021819360267459, 9.232143552801635, 9.518392708760017, 10.13130116663805, 10.73166705753032, 10.93452381039203, 11.48410623286924, 12.05084540620439, 12.57557405790202, 13.08986900908663
