| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s − 13-s − 2·15-s + 2·17-s − 4·21-s − 25-s + 27-s + 10·29-s + 4·31-s − 4·33-s + 8·35-s + 2·37-s − 39-s + 6·41-s + 12·43-s − 2·45-s + 9·49-s + 2·51-s − 6·53-s + 8·55-s − 12·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s − 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.696·33-s + 1.35·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169113170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169113170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.930621621641526010746712878659, −7.931023545928475276091772588410, −7.67003217829809798664468493665, −6.67604151273759236767429208217, −5.95291436601865739755607782383, −4.81540810133598275603719750743, −3.96533764142044298313757246268, −3.05586023372140157939993175531, −2.57147832048594013735645489348, −0.63730475324058459042740279262,
0.63730475324058459042740279262, 2.57147832048594013735645489348, 3.05586023372140157939993175531, 3.96533764142044298313757246268, 4.81540810133598275603719750743, 5.95291436601865739755607782383, 6.67604151273759236767429208217, 7.67003217829809798664468493665, 7.931023545928475276091772588410, 8.930621621641526010746712878659
