L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (−0.309 − 0.951i)4-s + (2.30 − 1.67i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.927 − 2.85i)8-s + (−0.809 − 0.587i)9-s + 2.85·10-s + (3.04 + 1.31i)11-s + 0.999·12-s + (1 + 0.726i)13-s + (−0.309 + 0.951i)14-s + (0.881 + 2.71i)15-s + (0.809 − 0.587i)16-s + (−6.35 + 4.61i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (−0.154 − 0.475i)4-s + (1.03 − 0.750i)5-s + (−0.330 + 0.239i)6-s + (0.116 + 0.359i)7-s + (0.327 − 1.00i)8-s + (−0.269 − 0.195i)9-s + 0.902·10-s + (0.918 + 0.396i)11-s + 0.288·12-s + (0.277 + 0.201i)13-s + (−0.0825 + 0.254i)14-s + (0.227 + 0.700i)15-s + (0.202 − 0.146i)16-s + (−1.54 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71470 + 0.210782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71470 + 0.210782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-3.04 - 1.31i)T \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.30 + 1.67i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1 - 0.726i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (6.35 - 4.61i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 2.48i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 + (0.618 + 1.90i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.92 - 4.30i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.73 + 11.4i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.35 - 10.3i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + (-0.618 + 1.90i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (9.85 + 7.15i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.09 - 6.43i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + (7.23 - 5.25i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.61 + 4.97i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-11.3 - 8.22i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.38 - 1.73i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 1.09T + 89T^{2} \) |
| 97 | \( 1 + (-2.85 - 2.07i)T + (29.9 + 92.2i)T^{2} \) |
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
−12.48972887429319633577655075314, −11.23626264243062273084326604977, −10.13986841208313170647785051459, −9.335467096186680626667057474237, −8.662948293132480471486146819153, −6.63935676954067093196231002282, −5.98646079563706530886480801015, −4.93479727158539279636699920102, −4.11901078430066018675588223709, −1.73056415830013496428354135776,
2.01714381392967332276919030370, 3.26339151780591006862708227484, 4.68856583053984370700299607531, 6.09573841132526209340137691233, 6.90213157301671675177539843498, 8.162266324129877762837448689735, 9.284024200792980560008303355235, 10.51685121185990164977226866974, 11.41522354404494113986686358785, 12.10567145487359175367003994627