| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 3·9-s − 2·10-s + 2·11-s + 3·13-s − 4·16-s + 6·18-s + 2·20-s − 4·22-s + 23-s + 25-s − 6·26-s − 8·29-s + 9·31-s + 8·32-s − 6·36-s − 3·37-s + 3·41-s + 6·43-s + 4·44-s − 3·45-s − 2·46-s − 3·47-s − 2·50-s + 6·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 9-s − 0.632·10-s + 0.603·11-s + 0.832·13-s − 16-s + 1.41·18-s + 0.447·20-s − 0.852·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 1.48·29-s + 1.61·31-s + 1.41·32-s − 36-s − 0.493·37-s + 0.468·41-s + 0.914·43-s + 0.603·44-s − 0.447·45-s − 0.294·46-s − 0.437·47-s − 0.282·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.21034316827504, −13.98057464647486, −13.49100707771924, −12.86321468713443, −12.26882536740560, −11.50400924422962, −11.31089535876312, −10.77342259931111, −10.30108379372721, −9.637747513791596, −9.262500311613480, −8.851530376513337, −8.380990096508142, −7.883339323489255, −7.355540862255739, −6.586144405533126, −6.265649735564542, −5.661560785353069, −4.989650443100574, −4.248428675241142, −3.568296597284640, −2.822228341186323, −2.183603562691694, −1.459930965185880, −0.8815928450515981, 0,
0.8815928450515981, 1.459930965185880, 2.183603562691694, 2.822228341186323, 3.568296597284640, 4.248428675241142, 4.989650443100574, 5.661560785353069, 6.265649735564542, 6.586144405533126, 7.355540862255739, 7.883339323489255, 8.380990096508142, 8.851530376513337, 9.262500311613480, 9.637747513791596, 10.30108379372721, 10.77342259931111, 11.31089535876312, 11.50400924422962, 12.26882536740560, 12.86321468713443, 13.49100707771924, 13.98057464647486, 14.21034316827504
