L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.118 − 2.23i)5-s + (−0.499 − 0.866i)6-s + 2.79i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.01 − 1.99i)10-s + 4.02·11-s − 0.999i·12-s + (0.0960 − 0.0554i)13-s + (−1.39 + 2.42i)14-s + (−1.01 + 1.99i)15-s + (−0.5 + 0.866i)16-s + (3.68 + 2.12i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.0531 − 0.998i)5-s + (−0.204 − 0.353i)6-s + 1.05i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.320 − 0.630i)10-s + 1.21·11-s − 0.288i·12-s + (0.0266 − 0.0153i)13-s + (−0.373 + 0.647i)14-s + (−0.261 + 0.514i)15-s + (−0.125 + 0.216i)16-s + (0.893 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88523 + 0.216199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88523 + 0.216199i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.118 + 2.23i)T \) |
| 19 | \( 1 + (-0.163 + 4.35i)T \) |
good | 7 | \( 1 - 2.79iT - 7T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 + (-0.0960 + 0.0554i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.68 - 2.12i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-7.65 + 4.42i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.907 - 1.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 - 1.68iT - 37T^{2} \) |
| 41 | \( 1 + (3.88 - 6.72i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.46 + 3.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.17 - 2.40i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.45 - 3.72i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.188 - 0.326i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.18 + 10.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.82 - 2.21i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.90 - 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.33 - 4.81i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.11 - 3.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.25iT - 83T^{2} \) |
| 89 | \( 1 + (-5.01 - 8.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.52 + 2.61i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.16679826911737388758869488529, −9.739902261063387151350436698518, −8.809493324568664771916502024539, −8.219302451891734694762857945740, −6.85858502647041137181352350009, −6.15925335506142758778176162938, −5.16042885154862421967675080139, −4.51489472088351161727381127967, −3.00440261228411149434321571303, −1.33117195892030511643172331287,
1.28627081245362057779957284341, 3.21705453215433387644003661503, 3.86095180823226042650136253781, 4.99366976425872956300024037353, 6.19387037856450769913303118099, 6.87908868001913821517042770345, 7.72289148495350450014811334687, 9.353663556196018757250292132345, 10.15404104525744504051311352365, 10.71625828599372864566531545833