| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)6-s + (−2.28 + 3.96i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.300i)11-s + (0.499 − 0.866i)12-s + (−4.99 − 4.18i)13-s + (4.29 + 1.56i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (−1.22 − 6.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.442 + 0.371i)3-s + (−0.469 + 0.171i)4-s + (0.420 + 0.152i)5-s + (0.312 + 0.262i)6-s + (−0.864 + 1.49i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (0.0523 + 0.0906i)11-s + (0.144 − 0.249i)12-s + (−1.38 − 1.16i)13-s + (1.14 + 0.418i)14-s + (−0.242 + 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.297 − 1.68i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00178958 + 0.0236006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00178958 + 0.0236006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (2.82 + 3.31i)T \) |
| good | 7 | \( 1 + (2.28 - 3.96i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.173 - 0.300i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.99 + 4.18i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.22 + 6.95i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (5.54 - 2.01i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.879 - 4.98i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.18 + 2.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + (-0.364 + 0.305i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.71 + 2.44i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.01 - 5.73i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (9.70 - 3.53i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.316 - 1.79i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (10.8 - 3.93i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.106 + 0.601i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 4.02i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.94 - 4.14i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.69 + 5.61i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.837 - 1.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.30 + 6.96i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.184 - 1.04i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.10691229556410284340485056783, −9.555268347356555853538471116185, −9.006215257634045046451807155839, −7.67275708947451469928999157253, −6.43880766910143723223214357833, −5.47694967120535716531199448696, −4.75151752269083759825676487710, −3.02261030040728011136244267223, −2.43528869449694980255672504847, −0.01381130773038887264942458046,
1.83766894431369852570530587819, 3.90898589491712851559781885559, 4.64753387090124779296856441098, 6.28427279014351557196320234106, 6.41587858156886046792002206998, 7.51670491658836863696205207568, 8.304651590376721489105213517953, 9.727860138125123902313309005018, 10.03398674925944294847004531973, 11.00120699308803479545398126297
