| L(s) = 1 | + (−1.40 + 0.124i)2-s + (0.673 − 1.59i)3-s + (1.96 − 0.350i)4-s + (−2.20 − 0.380i)5-s + (−0.750 + 2.33i)6-s − 0.0822i·7-s + (−2.73 + 0.738i)8-s + (−2.09 − 2.15i)9-s + (3.15 + 0.262i)10-s + (2.26 − 1.64i)11-s + (0.767 − 3.37i)12-s + (−3.85 − 2.80i)13-s + (0.0102 + 0.115i)14-s + (−2.09 + 3.25i)15-s + (3.75 − 1.38i)16-s + (−3.83 + 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0879i)2-s + (0.389 − 0.921i)3-s + (0.984 − 0.175i)4-s + (−0.985 − 0.170i)5-s + (−0.306 + 0.951i)6-s − 0.0310i·7-s + (−0.965 + 0.261i)8-s + (−0.697 − 0.716i)9-s + (0.996 + 0.0829i)10-s + (0.681 − 0.495i)11-s + (0.221 − 0.975i)12-s + (−1.06 − 0.777i)13-s + (0.00273 + 0.0309i)14-s + (−0.540 + 0.841i)15-s + (0.938 − 0.345i)16-s + (−0.929 + 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.126563 - 0.493019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.126563 - 0.493019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.124i)T \) |
| 3 | \( 1 + (-0.673 + 1.59i)T \) |
| 5 | \( 1 + (2.20 + 0.380i)T \) |
| good | 7 | \( 1 + 0.0822iT - 7T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.64i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.85 + 2.80i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.83 - 1.24i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.03 - 0.661i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.93 + 1.40i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.40 + 1.10i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.91 - 0.947i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.33 + 5.32i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.77 + 6.57i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.76iT - 43T^{2} \) |
| 47 | \( 1 + (0.650 - 2.00i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.15 - 2.32i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.56 - 5.49i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.43 - 1.77i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-12.2 + 3.97i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.11 - 3.42i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.23 - 0.898i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.51 - 1.46i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.96 + 9.13i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (6.21 + 8.55i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.56 - 7.89i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.35053941236649443328572146609, −10.47223684677714996102379573647, −8.959588464879243143941289198041, −8.611270612102872289630664286740, −7.44589376233658371170345756570, −7.00070254554074819150445565630, −5.67542136336960338251735652846, −3.69163495079689835454764331250, −2.27342445937183227575902112293, −0.46527630747746996779772061968,
2.35641972395770793707857721704, 3.73013126091936933632339345087, 4.82090203492128948348045551668, 6.69099674810034407425224312551, 7.48418524888425240594646369330, 8.594506295360434304199902215422, 9.280362785872823491136753377331, 10.08777136877338130680520858767, 11.22574929435719576270463864295, 11.59750099793273266847625389001
