| L(s) = 1 | + (0.746 − 1.04i)2-s + (0.235 − 0.971i)3-s + (0.112 + 0.324i)4-s + (0.116 − 2.45i)5-s + (−0.842 − 0.972i)6-s + (−2.38 − 1.13i)7-s + (2.89 + 0.849i)8-s + (−0.888 − 0.458i)9-s + (−2.48 − 1.95i)10-s + (−0.181 − 0.254i)11-s + (0.341 − 0.0326i)12-s + (0.439 − 3.05i)13-s + (−2.97 + 1.65i)14-s + (−2.35 − 0.692i)15-s + (2.51 − 1.97i)16-s + (2.40 − 0.463i)17-s + ⋯ |
| L(s) = 1 | + (0.527 − 0.741i)2-s + (0.136 − 0.561i)3-s + (0.0561 + 0.162i)4-s + (0.0522 − 1.09i)5-s + (−0.344 − 0.397i)6-s + (−0.902 − 0.429i)7-s + (1.02 + 0.300i)8-s + (−0.296 − 0.152i)9-s + (−0.785 − 0.618i)10-s + (−0.0546 − 0.0767i)11-s + (0.0986 − 0.00942i)12-s + (0.121 − 0.847i)13-s + (−0.795 + 0.442i)14-s + (−0.608 − 0.178i)15-s + (0.627 − 0.493i)16-s + (0.583 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759475 - 1.68718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759475 - 1.68718i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (2.38 + 1.13i)T \) |
| 23 | \( 1 + (4.70 + 0.916i)T \) |
| good | 2 | \( 1 + (-0.746 + 1.04i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.116 + 2.45i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (0.181 + 0.254i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.439 + 3.05i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 0.463i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (3.89 + 0.749i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-5.95 - 6.86i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-3.45 + 3.29i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (1.40 + 0.723i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-1.74 - 1.12i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-5.16 + 1.51i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (1.62 + 2.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.02 + 0.408i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-6.34 - 4.99i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (-0.958 - 3.95i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (-2.74 - 0.262i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (-1.70 - 3.73i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.47 + 4.25i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (12.8 - 5.13i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-0.742 + 0.477i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 9.96i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-2.31 - 1.48i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.70857175242384115086638765811, −10.00243001283417824451870907275, −8.735670639337681490312634802031, −8.045160138433450647761690724589, −7.02395294058985791524739069590, −5.84474296730152669181061487976, −4.67235620572912466056619786904, −3.62493937467174275596257262794, −2.54712657364289016070182406475, −0.982620869616578393267583667721,
2.35288801928695085566436936777, 3.63104904720846814129054346174, 4.65842461945894452275341525890, 6.07882157889448938103013931954, 6.34737795960743462076567312301, 7.37981152176377502365298578812, 8.546909741750110208634466203167, 9.866329245681581640213469534705, 10.18119869420925153976514714278, 11.16302494423559183020480427894
