| L(s) = 1 | − 5-s − 3·7-s − 2·11-s − 13-s + 6·17-s + 19-s + 2·23-s + 25-s + 9·29-s − 9·31-s + 3·35-s − 10·37-s + 3·41-s + 43-s − 4·47-s + 2·49-s − 10·53-s + 2·55-s − 8·59-s + 13·61-s + 65-s + 11·67-s + 8·71-s − 5·73-s + 6·77-s − 11·79-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 1.67·29-s − 1.61·31-s + 0.507·35-s − 1.64·37-s + 0.468·41-s + 0.152·43-s − 0.583·47-s + 2/7·49-s − 1.37·53-s + 0.269·55-s − 1.04·59-s + 1.66·61-s + 0.124·65-s + 1.34·67-s + 0.949·71-s − 0.585·73-s + 0.683·77-s − 1.23·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.64806089030713, −14.71127573785837, −14.35547967643025, −13.86643173692330, −13.07195642196075, −12.60589955083477, −12.41382664929678, −11.69586517956726, −11.08953197914951, −10.43706537189905, −9.993063362177687, −9.557109516173057, −8.867405941848812, −8.256059280185571, −7.686920287713852, −7.102708424886896, −6.644691155329125, −5.878418433012511, −5.294876066182259, −4.772196190632634, −3.823161389423909, −3.239022179464096, −2.917473355025254, −1.870558186348379, −0.8629682775274089, 0,
0.8629682775274089, 1.870558186348379, 2.917473355025254, 3.239022179464096, 3.823161389423909, 4.772196190632634, 5.294876066182259, 5.878418433012511, 6.644691155329125, 7.102708424886896, 7.686920287713852, 8.256059280185571, 8.867405941848812, 9.557109516173057, 9.993063362177687, 10.43706537189905, 11.08953197914951, 11.69586517956726, 12.41382664929678, 12.60589955083477, 13.07195642196075, 13.86643173692330, 14.35547967643025, 14.71127573785837, 15.64806089030713
