| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s − 11-s + 13-s − 3·15-s − 3·17-s − 8·19-s − 3·21-s + 4·23-s + 25-s − 9·27-s − 3·29-s − 6·31-s + 3·33-s + 35-s + 8·37-s − 3·39-s + 10·41-s + 12·43-s + 6·45-s + 3·47-s + 49-s + 9·51-s − 12·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.301·11-s + 0.277·13-s − 0.774·15-s − 0.727·17-s − 1.83·19-s − 0.654·21-s + 0.834·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s − 1.07·31-s + 0.522·33-s + 0.169·35-s + 1.31·37-s − 0.480·39-s + 1.56·41-s + 1.82·43-s + 0.894·45-s + 0.437·47-s + 1/7·49-s + 1.26·51-s − 1.64·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.798309551708936292993575984127, −7.63516401216804364260597262946, −6.92056654438047778543183362560, −5.97297029832093903453243196789, −5.80019968163880922615291196400, −4.62050138774362523714988503018, −4.23688965833285350273717678763, −2.48278256056882684163648729820, −1.33680268336365223138815805796, 0,
1.33680268336365223138815805796, 2.48278256056882684163648729820, 4.23688965833285350273717678763, 4.62050138774362523714988503018, 5.80019968163880922615291196400, 5.97297029832093903453243196789, 6.92056654438047778543183362560, 7.63516401216804364260597262946, 8.798309551708936292993575984127
