| L(s) = 1 | − 2·5-s − 7-s − 4·11-s + 6·13-s + 2·17-s − 4·19-s + 4·23-s − 25-s − 2·29-s + 8·31-s + 2·35-s + 10·37-s + 2·41-s − 8·43-s + 49-s − 10·53-s + 8·55-s − 12·59-s − 10·61-s − 12·65-s + 8·67-s − 12·71-s + 2·73-s + 4·77-s + 12·83-s − 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s + 1.64·37-s + 0.312·41-s − 1.21·43-s + 1/7·49-s − 1.37·53-s + 1.07·55-s − 1.56·59-s − 1.28·61-s − 1.48·65-s + 0.977·67-s − 1.42·71-s + 0.234·73-s + 0.455·77-s + 1.31·83-s − 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.011181072036533365936419098509, −7.60792077811478938052329030451, −6.44866980248976275785768350920, −6.04906526316956776040530428022, −4.97447762160079056191220656924, −4.21583069596413186654858023802, −3.39249646960069074957928538402, −2.69553050173518952505456691851, −1.27117621364021117770065652083, 0,
1.27117621364021117770065652083, 2.69553050173518952505456691851, 3.39249646960069074957928538402, 4.21583069596413186654858023802, 4.97447762160079056191220656924, 6.04906526316956776040530428022, 6.44866980248976275785768350920, 7.60792077811478938052329030451, 8.011181072036533365936419098509
