| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 6·13-s − 15-s − 2·17-s − 21-s − 8·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 33-s + 35-s + 10·37-s + 6·39-s − 10·41-s + 4·43-s + 45-s + 12·47-s + 49-s + 2·51-s + 2·53-s + 55-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 0.169·35-s + 1.64·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.134·55-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.11772764716990, −15.40490793322203, −15.00962480255129, −14.37081475899771, −13.88978831180372, −13.32155840641256, −12.63004934009470, −12.09736211294632, −11.70049574767543, −11.15049342389542, −10.30181043346247, −9.992500077355078, −9.467965487955447, −8.755952880071700, −7.981487734472800, −7.458542900386258, −6.846125788184389, −6.127983275768847, −5.654600447693690, −4.890699020393795, −4.411697687077024, −3.699535949346080, −2.413089950254865, −2.190283842645307, −1.034365262192448, 0,
1.034365262192448, 2.190283842645307, 2.413089950254865, 3.699535949346080, 4.411697687077024, 4.890699020393795, 5.654600447693690, 6.127983275768847, 6.846125788184389, 7.458542900386258, 7.981487734472800, 8.755952880071700, 9.467965487955447, 9.992500077355078, 10.30181043346247, 11.15049342389542, 11.70049574767543, 12.09736211294632, 12.63004934009470, 13.32155840641256, 13.88978831180372, 14.37081475899771, 15.00962480255129, 15.40490793322203, 16.11772764716990
