| L(s) = 1 | + (−0.120 − 0.162i)2-s + (1.25 + 1.19i)3-s + (0.561 − 1.87i)4-s + (−0.761 + 0.500i)5-s + (0.0416 − 0.347i)6-s + (−0.112 + 0.119i)7-s + (−0.751 + 0.273i)8-s + (0.157 + 2.99i)9-s + (0.173 + 0.0629i)10-s + (−0.670 + 0.336i)11-s + (2.94 − 1.68i)12-s + (−1.39 − 3.24i)13-s + (0.0329 + 0.00384i)14-s + (−1.55 − 0.278i)15-s + (−3.13 − 2.06i)16-s + (−3.09 + 2.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.0853 − 0.114i)2-s + (0.725 + 0.688i)3-s + (0.280 − 0.938i)4-s + (−0.340 + 0.223i)5-s + (0.0169 − 0.141i)6-s + (−0.0425 + 0.0450i)7-s + (−0.265 + 0.0967i)8-s + (0.0524 + 0.998i)9-s + (0.0547 + 0.0199i)10-s + (−0.202 + 0.101i)11-s + (0.849 − 0.487i)12-s + (−0.388 − 0.899i)13-s + (0.00880 + 0.00102i)14-s + (−0.401 − 0.0719i)15-s + (−0.784 − 0.516i)16-s + (−0.750 + 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07549 + 0.0147450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07549 + 0.0147450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.25 - 1.19i)T \) |
| good | 2 | \( 1 + (0.120 + 0.162i)T + (-0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (0.761 - 0.500i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (0.112 - 0.119i)T + (-0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (0.670 - 0.336i)T + (6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (1.39 + 3.24i)T + (-8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (3.09 - 2.59i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 1.36i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.63 + 5.96i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-6.53 + 0.764i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-5.44 + 1.29i)T + (27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-0.783 - 4.44i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-6.53 + 8.77i)T + (-11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.201 + 3.45i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-8.68 - 2.05i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-3.06 - 5.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.60 + 1.30i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-2.02 - 6.76i)T + (-50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (8.82 + 1.03i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (8.94 + 3.25i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.28 + 3.01i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.92 + 2.58i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (10.0 + 13.4i)T + (-23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (-10.7 + 3.90i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.78 - 2.48i)T + (38.4 + 89.0i)T^{2} \) |
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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
−14.54202621347378310078031922838, −13.62499803963827728161268262180, −12.09800973027342978107679396463, −10.62951194193598245039934048002, −10.19669066052682829067885753016, −8.883316855562209275951194083489, −7.64146827422871888436675335801, −5.94832368532595159747730829884, −4.43783977964379842579309123150, −2.58868131087763244157985561650,
2.57247655604177086434817743994, 4.16121623921910608745808208292, 6.56112595152419109879396174986, 7.56969611432325607599127502523, 8.467615098603328571456154563451, 9.567799733943535779282997846195, 11.58727059205510376726750058096, 12.16247360614857325314159306010, 13.34911382365756761618494534035, 14.10857738940096067996922742084
