| L(s) = 1 | + 2·2-s + 4·4-s + 13·7-s + 8·8-s + 30·11-s + 61·13-s + 26·14-s + 16·16-s + 12·17-s − 49·19-s + 60·22-s + 18·23-s + 122·26-s + 52·28-s + 186·29-s − 160·31-s + 32·32-s + 24·34-s + 91·37-s − 98·38-s − 378·41-s + 268·43-s + 120·44-s + 36·46-s + 144·47-s − 174·49-s + 244·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.701·7-s + 0.353·8-s + 0.822·11-s + 1.30·13-s + 0.496·14-s + 1/4·16-s + 0.171·17-s − 0.591·19-s + 0.581·22-s + 0.163·23-s + 0.920·26-s + 0.350·28-s + 1.19·29-s − 0.926·31-s + 0.176·32-s + 0.121·34-s + 0.404·37-s − 0.418·38-s − 1.43·41-s + 0.950·43-s + 0.411·44-s + 0.115·46-s + 0.446·47-s − 0.507·49-s + 0.650·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.483978909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.483978909\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 61 T + p^{3} T^{2} \) |
| 17 | \( 1 - 12 T + p^{3} T^{2} \) |
| 19 | \( 1 + 49 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 91 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 570 T + p^{3} T^{2} \) |
| 59 | \( 1 + 204 T + p^{3} T^{2} \) |
| 61 | \( 1 + 877 T + p^{3} T^{2} \) |
| 67 | \( 1 - 187 T + p^{3} T^{2} \) |
| 71 | \( 1 - 606 T + p^{3} T^{2} \) |
| 73 | \( 1 + 431 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1151 T + p^{3} T^{2} \) |
| 83 | \( 1 - 102 T + p^{3} T^{2} \) |
| 89 | \( 1 + 984 T + p^{3} T^{2} \) |
| 97 | \( 1 - 265 T + p^{3} T^{2} \) |
| 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
−9.050759086506745187315203018032, −8.460334729663640578664673420094, −7.53643885401769296203723899137, −6.56091886698993112303441187446, −5.95492641018148902750910356404, −4.93123566445431547962795378985, −4.10883607887356400661898750078, −3.30639325060974933467038329242, −1.98144271771846333092703159871, −1.02988075410476743991441014115,
1.02988075410476743991441014115, 1.98144271771846333092703159871, 3.30639325060974933467038329242, 4.10883607887356400661898750078, 4.93123566445431547962795378985, 5.95492641018148902750910356404, 6.56091886698993112303441187446, 7.53643885401769296203723899137, 8.460334729663640578664673420094, 9.050759086506745187315203018032
