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
−11.00120699308803479545398126297, −10.03398674925944294847004531973, −9.727860138125123902313309005018, −8.304651590376721489105213517953, −7.51670491658836863696205207568, −6.41587858156886046792002206998, −6.28427279014351557196320234106, −4.64753387090124779296856441098, −3.90898589491712851559781885559, −1.83766894431369852570530587819,
0.01381130773038887264942458046, 2.43528869449694980255672504847, 3.02261030040728011136244267223, 4.75151752269083759825676487710, 5.47694967120535716531199448696, 6.43880766910143723223214357833, 7.67275708947451469928999157253, 9.006215257634045046451807155839, 9.555268347356555853538471116185, 10.10691229556410284340485056783