| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s + 5·7-s + 8·8-s − 18·9-s − 4·10-s − 13·11-s − 12·12-s + 10·14-s + 6·15-s + 16·16-s + 27·17-s − 36·18-s − 75·19-s − 8·20-s − 15·21-s − 26·22-s − 187·23-s − 24·24-s − 121·25-s + 135·27-s + 20·28-s − 13·29-s + 12·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.178·5-s − 0.408·6-s + 0.269·7-s + 0.353·8-s − 2/3·9-s − 0.126·10-s − 0.356·11-s − 0.288·12-s + 0.190·14-s + 0.103·15-s + 1/4·16-s + 0.385·17-s − 0.471·18-s − 0.905·19-s − 0.0894·20-s − 0.155·21-s − 0.251·22-s − 1.69·23-s − 0.204·24-s − 0.967·25-s + 0.962·27-s + 0.134·28-s − 0.0832·29-s + 0.0730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p^{3} T^{2} \) |
| 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 13 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 + 75 T + p^{3} T^{2} \) |
| 23 | \( 1 + 187 T + p^{3} T^{2} \) |
| 29 | \( 1 + 13 T + p^{3} T^{2} \) |
| 31 | \( 1 - 104 T + p^{3} T^{2} \) |
| 37 | \( 1 + 423 T + p^{3} T^{2} \) |
| 41 | \( 1 + 195 T + p^{3} T^{2} \) |
| 43 | \( 1 - 199 T + p^{3} T^{2} \) |
| 47 | \( 1 + 388 T + p^{3} T^{2} \) |
| 53 | \( 1 - 618 T + p^{3} T^{2} \) |
| 59 | \( 1 + 491 T + p^{3} T^{2} \) |
| 61 | \( 1 - 175 T + p^{3} T^{2} \) |
| 67 | \( 1 + 817 T + p^{3} T^{2} \) |
| 71 | \( 1 + 79 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 764 T + p^{3} T^{2} \) |
| 83 | \( 1 - 732 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1041 T + p^{3} T^{2} \) |
| 97 | \( 1 - p T + p^{3} T^{2} \) |
| 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
−10.80626238979821122878367944514, −10.07962631059330393675863614463, −8.570757296948365112386790746908, −7.73977360828223591662600357142, −6.43865235123607295894654563151, −5.68203344292140942932582325364, −4.69063421333706843351600023443, −3.49121260591226961096832690557, −2.02766766009684016505747627745, 0,
2.02766766009684016505747627745, 3.49121260591226961096832690557, 4.69063421333706843351600023443, 5.68203344292140942932582325364, 6.43865235123607295894654563151, 7.73977360828223591662600357142, 8.570757296948365112386790746908, 10.07962631059330393675863614463, 10.80626238979821122878367944514
