| L(s) = 1 | − 0.456·2-s − 1.79·4-s − 4.37·5-s − 7-s + 1.73·8-s + 1.99·10-s − 2.64·11-s + 4·13-s + 0.456·14-s + 2.79·16-s + 3.46·17-s − 3.58·19-s + 7.84·20-s + 1.20·22-s − 3.46·23-s + 14.1·25-s − 1.82·26-s + 1.79·28-s + 1.82·29-s + 9.16·31-s − 4.73·32-s − 1.58·34-s + 4.37·35-s + 3·37-s + 1.63·38-s − 7.58·40-s + 4.37·41-s + ⋯ |
| L(s) = 1 | − 0.323·2-s − 0.895·4-s − 1.95·5-s − 0.377·7-s + 0.612·8-s + 0.632·10-s − 0.797·11-s + 1.10·13-s + 0.122·14-s + 0.697·16-s + 0.840·17-s − 0.821·19-s + 1.75·20-s + 0.257·22-s − 0.722·23-s + 2.83·25-s − 0.358·26-s + 0.338·28-s + 0.339·29-s + 1.64·31-s − 0.837·32-s − 0.271·34-s + 0.739·35-s + 0.493·37-s + 0.265·38-s − 1.19·40-s + 0.683·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5304539060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5304539060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 0.456T + 2T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.82T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 + 8.58T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 - 8.75T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69046576984660308297900147317, −9.943032551241114955340318596669, −8.633661634193135992434590262352, −8.201420589479745284105441690784, −7.56782670746839869021950662367, −6.27590427137137293127582129876, −4.85957993614369296498150202078, −4.04238941355116605697051178395, −3.21923382511238821807819438005, −0.67478738318070858054725244698,
0.67478738318070858054725244698, 3.21923382511238821807819438005, 4.04238941355116605697051178395, 4.85957993614369296498150202078, 6.27590427137137293127582129876, 7.56782670746839869021950662367, 8.201420589479745284105441690784, 8.633661634193135992434590262352, 9.943032551241114955340318596669, 10.69046576984660308297900147317
