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 | \( 1 + (-2.85 + 2.07i)T + (29.9 - 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10567145487359175367003994627, −11.41522354404494113986686358785, −10.51685121185990164977226866974, −9.284024200792980560008303355235, −8.162266324129877762837448689735, −6.90213157301671675177539843498, −6.09573841132526209340137691233, −4.68856583053984370700299607531, −3.26339151780591006862708227484, −2.01714381392967332276919030370,
1.73056415830013496428354135776, 4.11901078430066018675588223709, 4.93479727158539279636699920102, 5.98646079563706530886480801015, 6.63935676954067093196231002282, 8.662948293132480471486146819153, 9.335467096186680626667057474237, 10.13986841208313170647785051459, 11.23626264243062273084326604977, 12.48972887429319633577655075314