| L(s) = 1 | + 4·2-s − 6·3-s + 16·4-s + 25·5-s − 24·6-s + 64·8-s − 207·9-s + 100·10-s + 192·11-s − 96·12-s − 1.10e3·13-s − 150·15-s + 256·16-s − 762·17-s − 828·18-s + 2.74e3·19-s + 400·20-s + 768·22-s + 1.56e3·23-s − 384·24-s + 625·25-s − 4.42e3·26-s + 2.70e3·27-s + 5.91e3·29-s − 600·30-s + 6.86e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.447·5-s − 0.272·6-s + 0.353·8-s − 0.851·9-s + 0.316·10-s + 0.478·11-s − 0.192·12-s − 1.81·13-s − 0.172·15-s + 1/4·16-s − 0.639·17-s − 0.602·18-s + 1.74·19-s + 0.223·20-s + 0.338·22-s + 0.617·23-s − 0.136·24-s + 1/5·25-s − 1.28·26-s + 0.712·27-s + 1.30·29-s − 0.121·30-s + 1.28·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.924741026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.924741026\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 192 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 + 762 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1566 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5910 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6868 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5518 T + p^{5} T^{2} \) |
| 41 | \( 1 - 378 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2434 T + p^{5} T^{2} \) |
| 47 | \( 1 + 13122 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9174 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 9838 T + p^{5} T^{2} \) |
| 67 | \( 1 - 33722 T + p^{5} T^{2} \) |
| 71 | \( 1 - 70212 T + p^{5} T^{2} \) |
| 73 | \( 1 + 21986 T + p^{5} T^{2} \) |
| 79 | \( 1 - 4520 T + p^{5} T^{2} \) |
| 83 | \( 1 - 109074 T + p^{5} T^{2} \) |
| 89 | \( 1 + 38490 T + p^{5} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{5} T^{2} \) |
| 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
−10.17409215000499471586758442478, −9.481011684254423336584831630925, −8.302834804423623641737349389614, −7.11453114774098521024056633626, −6.40741687381646647441797904755, −5.23828599299800380088512640450, −4.80222254379457634718478353231, −3.19746780888978486422054544797, −2.34659589496063485168822440313, −0.78573878838011113200775415555,
0.78573878838011113200775415555, 2.34659589496063485168822440313, 3.19746780888978486422054544797, 4.80222254379457634718478353231, 5.23828599299800380088512640450, 6.40741687381646647441797904755, 7.11453114774098521024056633626, 8.302834804423623641737349389614, 9.481011684254423336584831630925, 10.17409215000499471586758442478
