| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 6·17-s + 19-s + 21-s − 8·23-s − 4·25-s − 27-s − 7·29-s − 10·31-s − 33-s + 35-s − 3·37-s + 39-s − 4·43-s − 45-s + 7·47-s + 49-s − 6·51-s + 4·53-s − 55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.229·19-s + 0.218·21-s − 1.66·23-s − 4/5·25-s − 0.192·27-s − 1.29·29-s − 1.79·31-s − 0.174·33-s + 0.169·35-s − 0.493·37-s + 0.160·39-s − 0.609·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s − 0.134·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.807474772744232331134009383976, −8.919905883891253125858165855216, −7.68673904619298580477462490881, −7.30319789233421970761044838265, −5.98052261658296462479834897331, −5.50518636446989610618898632627, −4.14932920062052701879103763967, −3.41714013078349444399543761452, −1.75409205903664090661478855282, 0,
1.75409205903664090661478855282, 3.41714013078349444399543761452, 4.14932920062052701879103763967, 5.50518636446989610618898632627, 5.98052261658296462479834897331, 7.30319789233421970761044838265, 7.68673904619298580477462490881, 8.919905883891253125858165855216, 9.807474772744232331134009383976
