| L(s) = 1 | − 2-s − 3·3-s + 4-s − 2·5-s + 3·6-s − 8-s + 6·9-s + 2·10-s + 2·11-s − 3·12-s + 6·13-s + 6·15-s + 16-s − 6·18-s − 7·19-s − 2·20-s − 2·22-s + 8·23-s + 3·24-s − 25-s − 6·26-s − 9·27-s + 9·29-s − 6·30-s + 5·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s + 1.22·6-s − 0.353·8-s + 2·9-s + 0.632·10-s + 0.603·11-s − 0.866·12-s + 1.66·13-s + 1.54·15-s + 1/4·16-s − 1.41·18-s − 1.60·19-s − 0.447·20-s − 0.426·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s − 1.17·26-s − 1.73·27-s + 1.67·29-s − 1.09·30-s + 0.898·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8151260465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8151260465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.52654933274558, −15.10450113196876, −14.16932271833123, −13.40420357009362, −12.87810563862071, −12.28304457841746, −11.75600383552065, −11.50342776179002, −10.76147195105140, −10.67757138418042, −10.07995224407352, −9.104027753989200, −8.653426057880298, −8.148730885989938, −7.366345884802628, −6.663312634341825, −6.385450145092041, −5.983223034338680, −4.963869609527670, −4.528848872441919, −3.831834697190416, −3.128858563982343, −1.864193706476033, −1.011033963081360, −0.5720215677836166,
0.5720215677836166, 1.011033963081360, 1.864193706476033, 3.128858563982343, 3.831834697190416, 4.528848872441919, 4.963869609527670, 5.983223034338680, 6.385450145092041, 6.663312634341825, 7.366345884802628, 8.148730885989938, 8.653426057880298, 9.104027753989200, 10.07995224407352, 10.67757138418042, 10.76147195105140, 11.50342776179002, 11.75600383552065, 12.28304457841746, 12.87810563862071, 13.40420357009362, 14.16932271833123, 15.10450113196876, 15.52654933274558
