| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s + 2·13-s + 4·15-s + 2·17-s + 4·19-s − 2·21-s − 25-s + 4·27-s + 2·29-s + 10·31-s − 2·35-s + 8·37-s − 4·39-s + 2·41-s + 4·43-s − 2·45-s + 6·47-s + 49-s − 4·51-s − 12·53-s − 8·57-s + 10·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 0.485·17-s + 0.917·19-s − 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.79·31-s − 0.338·35-s + 1.31·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s − 1.64·53-s − 1.05·57-s + 1.30·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907104590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907104590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.81653971037072, −12.14984472436954, −11.93816887922578, −11.51392972097365, −11.19728391942426, −10.79442726967678, −10.12025299140415, −9.871731490695512, −9.154722527971073, −8.551911606608587, −8.181514957315934, −7.540809722153790, −7.421787956801257, −6.556226408482391, −6.101571030574674, −5.852818588892983, −5.107734623662331, −4.728640746625910, −4.250152229185730, −3.677202594608136, −3.025574702187503, −2.529841341805109, −1.538384446092606, −0.8263635785516721, −0.5925402483997291,
0.5925402483997291, 0.8263635785516721, 1.538384446092606, 2.529841341805109, 3.025574702187503, 3.677202594608136, 4.250152229185730, 4.728640746625910, 5.107734623662331, 5.852818588892983, 6.101571030574674, 6.556226408482391, 7.421787956801257, 7.540809722153790, 8.181514957315934, 8.551911606608587, 9.154722527971073, 9.871731490695512, 10.12025299140415, 10.79442726967678, 11.19728391942426, 11.51392972097365, 11.93816887922578, 12.14984472436954, 12.81653971037072
