L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·13-s − 2·15-s − 2·17-s − 4·19-s + 8·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 6·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s − 2·53-s − 4·57-s − 4·59-s + 2·61-s − 4·65-s + 4·67-s + 8·69-s − 8·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.63618059284105349520809602510, −7.32438284862293693328503541547, −6.42508206754472676439323141059, −5.62755823403878952826962811199, −4.60156021497859087443489366122, −4.01628213317951587527532774984, −3.32790226181780202269440904381, −2.43505247067401554863084291555, −1.37409591703931866594987960910, 0,
1.37409591703931866594987960910, 2.43505247067401554863084291555, 3.32790226181780202269440904381, 4.01628213317951587527532774984, 4.60156021497859087443489366122, 5.62755823403878952826962811199, 6.42508206754472676439323141059, 7.32438284862293693328503541547, 7.63618059284105349520809602510