| L(s) = 1 | − 2·5-s + 2·11-s + 4·13-s − 2·17-s + 2·19-s + 4·23-s − 25-s + 6·29-s + 6·31-s + 37-s + 6·41-s − 4·43-s + 4·53-s − 4·55-s + 4·59-s + 8·61-s − 8·65-s − 4·67-s + 6·71-s + 10·73-s + 8·79-s − 12·83-s + 4·85-s − 10·89-s − 4·95-s + 16·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.164·37-s + 0.937·41-s − 0.609·43-s + 0.549·53-s − 0.539·55-s + 0.520·59-s + 1.02·61-s − 0.992·65-s − 0.488·67-s + 0.712·71-s + 1.17·73-s + 0.900·79-s − 1.31·83-s + 0.433·85-s − 1.05·89-s − 0.410·95-s + 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.025528145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025528145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−13.61810933541096, −12.88041270606707, −12.60535410184084, −11.79252082724407, −11.65117207523826, −11.18957995700916, −10.70878116062625, −10.07507750966667, −9.636769353057979, −8.944568982757881, −8.546648313658465, −8.219123553059478, −7.560782193540225, −7.075774317846541, −6.499216436554617, −6.153706426451790, −5.443803684642895, −4.749457379151416, −4.335922353104224, −3.731208254996398, −3.322210128204098, −2.646170985003518, −1.888800110170785, −0.9659406873835599, −0.6731831791658929,
0.6731831791658929, 0.9659406873835599, 1.888800110170785, 2.646170985003518, 3.322210128204098, 3.731208254996398, 4.335922353104224, 4.749457379151416, 5.443803684642895, 6.153706426451790, 6.499216436554617, 7.075774317846541, 7.560782193540225, 8.219123553059478, 8.546648313658465, 8.944568982757881, 9.636769353057979, 10.07507750966667, 10.70878116062625, 11.18957995700916, 11.65117207523826, 11.79252082724407, 12.60535410184084, 12.88041270606707, 13.61810933541096
