| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s + 7-s + 9-s − 2·10-s − 11-s − 2·12-s − 2·13-s − 2·14-s − 15-s − 4·16-s − 3·17-s − 2·18-s + 19-s + 2·20-s − 21-s + 2·22-s − 7·23-s + 25-s + 4·26-s − 27-s + 2·28-s + 29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 0.258·15-s − 16-s − 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s − 0.218·21-s + 0.426·22-s − 1.45·23-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 0.185·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−9.580913403159408107746538839399, −8.525863371755079766633113323917, −7.969170568653515044214463691545, −7.02803943705769353382088003020, −6.28400126867765721727742786785, −5.16897274933370109786717977716, −4.30419949362677255982283333002, −2.50495161201074940955654386464, −1.47308385103957437581999459253, 0,
1.47308385103957437581999459253, 2.50495161201074940955654386464, 4.30419949362677255982283333002, 5.16897274933370109786717977716, 6.28400126867765721727742786785, 7.02803943705769353382088003020, 7.969170568653515044214463691545, 8.525863371755079766633113323917, 9.580913403159408107746538839399
