| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s − 2·13-s − 4·15-s + 6·17-s − 6·19-s + 2·21-s + 4·23-s − 25-s + 4·27-s − 8·31-s − 2·35-s − 8·37-s + 4·39-s + 6·41-s − 8·43-s + 2·45-s + 49-s − 12·51-s + 4·53-s + 12·57-s − 6·59-s + 2·61-s − 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.03·15-s + 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.43·31-s − 0.338·35-s − 1.31·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s + 1/7·49-s − 1.68·51-s + 0.549·53-s + 1.58·57-s − 0.781·59-s + 0.256·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.090852512564851802613788286292, −8.073847681736889055930928744294, −6.99246131051166828753464139127, −6.41485741375964366183296639351, −5.47867350522858162884171552310, −5.25891723488831478168816715334, −3.93857515171648208537257789369, −2.73639830864108581497225505140, −1.51014559356040138195013675701, 0,
1.51014559356040138195013675701, 2.73639830864108581497225505140, 3.93857515171648208537257789369, 5.25891723488831478168816715334, 5.47867350522858162884171552310, 6.41485741375964366183296639351, 6.99246131051166828753464139127, 8.073847681736889055930928744294, 9.090852512564851802613788286292
