| L(s) = 1 | − 3·7-s + 4·11-s + 7·13-s + 4·17-s + 19-s + 8·23-s + 3·31-s − 2·37-s + 6·41-s − 11·43-s − 6·47-s + 2·49-s + 6·53-s − 6·59-s − 61-s − 15·67-s + 6·71-s − 2·73-s − 12·77-s + 8·79-s − 2·83-s + 16·89-s − 21·91-s − 13·97-s + 2·101-s − 16·103-s + 18·107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 1.20·11-s + 1.94·13-s + 0.970·17-s + 0.229·19-s + 1.66·23-s + 0.538·31-s − 0.328·37-s + 0.937·41-s − 1.67·43-s − 0.875·47-s + 2/7·49-s + 0.824·53-s − 0.781·59-s − 0.128·61-s − 1.83·67-s + 0.712·71-s − 0.234·73-s − 1.36·77-s + 0.900·79-s − 0.219·83-s + 1.69·89-s − 2.20·91-s − 1.31·97-s + 0.199·101-s − 1.57·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.364204825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.364204825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| 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
−7.969305931097797509781291631259, −7.04028674300523642076775396574, −6.44977956101137224970043074790, −6.06390536720697176664637444583, −5.17671613422692599549380066866, −4.17149426050621873098984446071, −3.35741026901227201301276822653, −3.13446639095485905475807002765, −1.54746945513179656369988677208, −0.859971002084250164641308616391,
0.859971002084250164641308616391, 1.54746945513179656369988677208, 3.13446639095485905475807002765, 3.35741026901227201301276822653, 4.17149426050621873098984446071, 5.17671613422692599549380066866, 6.06390536720697176664637444583, 6.44977956101137224970043074790, 7.04028674300523642076775396574, 7.969305931097797509781291631259
