| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s − 2·13-s − 15-s + 2·17-s + 4·19-s − 21-s + 4·23-s + 25-s − 27-s + 6·29-s + 33-s + 35-s − 2·37-s + 2·39-s + 6·41-s + 12·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 6·53-s − 55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.005849520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.005849520\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.15326380080511, −13.50911763452392, −13.10006547590157, −12.48773235251554, −12.13973696086585, −11.55329521420495, −11.11792219495838, −10.52717317327045, −10.14830400288171, −9.519538276815772, −9.217493199950489, −8.371394409159768, −7.964952685377051, −7.262896590540895, −6.931453304552809, −6.257567470875487, −5.599421574196533, −5.203148157514202, −4.801674733843454, −4.051090565754362, −3.348600310400024, −2.587909471937020, −2.111031622593299, −1.047053687373121, −0.7262979870554005,
0.7262979870554005, 1.047053687373121, 2.111031622593299, 2.587909471937020, 3.348600310400024, 4.051090565754362, 4.801674733843454, 5.203148157514202, 5.599421574196533, 6.257567470875487, 6.931453304552809, 7.262896590540895, 7.964952685377051, 8.371394409159768, 9.217493199950489, 9.519538276815772, 10.14830400288171, 10.52717317327045, 11.11792219495838, 11.55329521420495, 12.13973696086585, 12.48773235251554, 13.10006547590157, 13.50911763452392, 14.15326380080511
