| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 8·17-s + 18-s + 2·19-s − 20-s + 21-s + 22-s + 24-s + 25-s + 26-s + 27-s + 28-s − 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.33928928769103, −12.96688984075510, −12.61711934300985, −12.02714865537121, −11.63264728600659, −11.11239306843959, −10.83623013183062, −9.977896856251311, −9.671516914428258, −9.241235299464078, −8.452979887632502, −8.074797443164712, −7.554296093690362, −7.360013436762980, −6.578081376124244, −6.079921548431255, −5.417785172995932, −5.100296425292471, −4.453998937286226, −3.804763440203629, −3.343300825034288, −3.161177218975968, −2.211705389982105, −1.536722664156161, −1.158236211582295, 0,
1.158236211582295, 1.536722664156161, 2.211705389982105, 3.161177218975968, 3.343300825034288, 3.804763440203629, 4.453998937286226, 5.100296425292471, 5.417785172995932, 6.079921548431255, 6.578081376124244, 7.360013436762980, 7.554296093690362, 8.074797443164712, 8.452979887632502, 9.241235299464078, 9.671516914428258, 9.977896856251311, 10.83623013183062, 11.11239306843959, 11.63264728600659, 12.02714865537121, 12.61711934300985, 12.96688984075510, 13.33928928769103
