L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 4·11-s − 6·13-s + 2·15-s − 2·17-s − 4·19-s − 21-s + 4·23-s − 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 2·35-s − 10·37-s + 6·39-s − 2·41-s − 8·43-s − 2·45-s + 49-s + 2·51-s − 10·53-s − 8·55-s + 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.298·45-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 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
−10.15558037915267119188449112735, −9.206574907669103417078639034320, −8.330869971877962461708394341895, −7.20214522889553119059062687435, −6.78044077445841975719051191645, −5.36857771747617040863256466681, −4.52969287471330622027217048773, −3.61417461907435968748262507791, −1.90612310330978239121510440566, 0,
1.90612310330978239121510440566, 3.61417461907435968748262507791, 4.52969287471330622027217048773, 5.36857771747617040863256466681, 6.78044077445841975719051191645, 7.20214522889553119059062687435, 8.330869971877962461708394341895, 9.206574907669103417078639034320, 10.15558037915267119188449112735