| L(s) = 1 | + 8.81·2-s − 28.7·3-s + 45.6·4-s − 253.·6-s + 98.9·7-s + 120.·8-s + 585.·9-s + 107.·11-s − 1.31e3·12-s + 345.·13-s + 872.·14-s − 398.·16-s + 1.40e3·17-s + 5.16e3·18-s − 138.·19-s − 2.84e3·21-s + 945.·22-s + 3.55e3·23-s − 3.47e3·24-s + 3.04e3·26-s − 9.86e3·27-s + 4.52e3·28-s − 1.51e3·29-s + 1.55e3·31-s − 7.37e3·32-s − 3.08e3·33-s + 1.23e4·34-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 1.84·3-s + 1.42·4-s − 2.87·6-s + 0.763·7-s + 0.666·8-s + 2.41·9-s + 0.267·11-s − 2.63·12-s + 0.566·13-s + 1.18·14-s − 0.388·16-s + 1.17·17-s + 3.75·18-s − 0.0878·19-s − 1.40·21-s + 0.416·22-s + 1.40·23-s − 1.23·24-s + 0.883·26-s − 2.60·27-s + 1.08·28-s − 0.334·29-s + 0.290·31-s − 1.27·32-s − 0.493·33-s + 1.83·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.734588606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734588606\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 8.81T + 32T^{2} \) |
| 3 | \( 1 + 28.7T + 243T^{2} \) |
| 7 | \( 1 - 98.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 107.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 345.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.40e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 138.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 192.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 129.T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.05e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.47e4T + 8.58e9T^{2} \) |
| 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
−9.401329590021959151500745580204, −7.964784286633080395154218176770, −6.88728141460430243755298831566, −6.35582818978335812711581278658, −5.41284929292076324377078345234, −5.10115200459963297137779089354, −4.26312142751561897248654100150, −3.31803940153881197177123731502, −1.69852578949970675711686851918, −0.76062057689844396826345111681,
0.76062057689844396826345111681, 1.69852578949970675711686851918, 3.31803940153881197177123731502, 4.26312142751561897248654100150, 5.10115200459963297137779089354, 5.41284929292076324377078345234, 6.35582818978335812711581278658, 6.88728141460430243755298831566, 7.964784286633080395154218176770, 9.401329590021959151500745580204
