| L(s) = 1 | + (−1.27 + 0.602i)2-s + (−0.900 − 0.433i)3-s + (1.27 − 1.54i)4-s + (0.0432 − 0.0897i)5-s + (1.41 + 0.0123i)6-s + (0.103 + 2.64i)7-s + (−0.701 + 2.74i)8-s + (0.623 + 0.781i)9-s + (−0.00122 + 0.140i)10-s + (−2.51 − 2.00i)11-s + (−1.81 + 0.836i)12-s + (0.338 + 0.270i)13-s + (−1.72 − 3.32i)14-s + (−0.0778 + 0.0620i)15-s + (−0.753 − 3.92i)16-s + (1.76 + 0.402i)17-s + ⋯ |
| L(s) = 1 | + (−0.904 + 0.426i)2-s + (−0.520 − 0.250i)3-s + (0.637 − 0.770i)4-s + (0.0193 − 0.0401i)5-s + (0.577 + 0.00503i)6-s + (0.0389 + 0.999i)7-s + (−0.247 + 0.968i)8-s + (0.207 + 0.260i)9-s + (−0.000388 + 0.0445i)10-s + (−0.759 − 0.605i)11-s + (−0.524 + 0.241i)12-s + (0.0940 + 0.0749i)13-s + (−0.460 − 0.887i)14-s + (−0.0201 + 0.0160i)15-s + (−0.188 − 0.982i)16-s + (0.427 + 0.0975i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.149814 + 0.385694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149814 + 0.385694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.602i)T \) |
| 3 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.103 - 2.64i)T \) |
| good | 5 | \( 1 + (-0.0432 + 0.0897i)T + (-3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (2.51 + 2.00i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.338 - 0.270i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.402i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 0.291T + 19T^{2} \) |
| 23 | \( 1 + (8.74 - 1.99i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.201 + 0.881i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 + (0.872 - 3.82i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (4.00 - 8.31i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.80 - 9.98i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (3.95 - 4.95i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (1.58 + 6.95i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (10.7 - 5.16i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.494 + 0.112i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 7.57iT - 67T^{2} \) |
| 71 | \( 1 + (-1.74 + 0.397i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (11.1 - 8.90i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + 4.06iT - 79T^{2} \) |
| 83 | \( 1 + (-9.81 - 12.3i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.04 + 2.43i)T + (19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 - 4.63iT - 97T^{2} \) |
| 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
−11.01584323994737036985091656185, −10.02604727162009595811331367189, −9.290247182142326799270596173004, −8.203966994732738950060420514009, −7.75496127297460548790120887295, −6.40539892969862097001822425352, −5.82347785683589071746957915207, −4.96816145171032160269455749884, −2.94804774907948527161899099913, −1.58511411606787282761977374206,
0.32709958994615070333186694984, 2.01246255430670849411630991979, 3.53719787302388508803731477670, 4.53981027962146609755287912243, 5.92427920531491977497179853107, 7.05899581302629262552864810411, 7.67760738378250147575092600262, 8.669712823971460141502484079139, 9.815145727217592192109121016421, 10.42458530866475094693818127946
