| L(s) = 1 | − 2·2-s − 8·3-s + 4·4-s + 16·6-s + 12·7-s − 8·8-s + 37·9-s − 11·11-s − 32·12-s + 34·13-s − 24·14-s + 16·16-s + 86·17-s − 74·18-s − 4·19-s − 96·21-s + 22·22-s − 148·23-s + 64·24-s − 68·26-s − 80·27-s + 48·28-s + 134·29-s − 280·31-s − 32·32-s + 88·33-s − 172·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s + 1/2·4-s + 1.08·6-s + 0.647·7-s − 0.353·8-s + 1.37·9-s − 0.301·11-s − 0.769·12-s + 0.725·13-s − 0.458·14-s + 1/4·16-s + 1.22·17-s − 0.968·18-s − 0.0482·19-s − 0.997·21-s + 0.213·22-s − 1.34·23-s + 0.544·24-s − 0.512·26-s − 0.570·27-s + 0.323·28-s + 0.858·29-s − 1.62·31-s − 0.176·32-s + 0.464·33-s − 0.867·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7475961414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7475961414\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 86 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 148 T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 280 T + p^{3} T^{2} \) |
| 37 | \( 1 + 430 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 136 T + p^{3} T^{2} \) |
| 47 | \( 1 - 28 T + p^{3} T^{2} \) |
| 53 | \( 1 - 658 T + p^{3} T^{2} \) |
| 59 | \( 1 - 4 T + p^{3} T^{2} \) |
| 61 | \( 1 + 90 T + p^{3} T^{2} \) |
| 67 | \( 1 + 96 T + p^{3} T^{2} \) |
| 71 | \( 1 - 816 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1296 T + p^{3} T^{2} \) |
| 83 | \( 1 - 608 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 706 T + p^{3} T^{2} \) |
| 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
−10.61415206474244891142749471340, −9.767413598032094930404801192997, −8.535091961393403692064782086623, −7.69156754779356092109138591212, −6.70382600435902750684005796377, −5.75732667084941940398999919901, −5.13416514457059256172568364424, −3.71333604975376104606532490722, −1.79727874061379903101303649161, −0.64658154572408946318879274176,
0.64658154572408946318879274176, 1.79727874061379903101303649161, 3.71333604975376104606532490722, 5.13416514457059256172568364424, 5.75732667084941940398999919901, 6.70382600435902750684005796377, 7.69156754779356092109138591212, 8.535091961393403692064782086623, 9.767413598032094930404801192997, 10.61415206474244891142749471340
