L(s) = 1 | + (−1.34 − 0.427i)2-s + (−0.281 − 1.86i)3-s + (1.63 + 1.15i)4-s + (2.47 + 0.970i)5-s + (−0.418 + 2.63i)6-s + (−2.31 − 1.28i)7-s + (−1.71 − 2.25i)8-s + (−0.546 + 0.168i)9-s + (−2.91 − 2.36i)10-s + (0.464 − 1.50i)11-s + (1.69 − 3.37i)12-s + (0.149 − 0.0341i)13-s + (2.56 + 2.72i)14-s + (1.11 − 4.89i)15-s + (1.34 + 3.76i)16-s + (0.505 − 0.344i)17-s + ⋯ |
L(s) = 1 | + (−0.953 − 0.302i)2-s + (−0.162 − 1.07i)3-s + (0.817 + 0.575i)4-s + (1.10 + 0.433i)5-s + (−0.170 + 1.07i)6-s + (−0.873 − 0.487i)7-s + (−0.605 − 0.795i)8-s + (−0.182 + 0.0561i)9-s + (−0.922 − 0.747i)10-s + (0.140 − 0.454i)11-s + (0.488 − 0.975i)12-s + (0.0414 − 0.00945i)13-s + (0.685 + 0.728i)14-s + (0.288 − 1.26i)15-s + (0.336 + 0.941i)16-s + (0.122 − 0.0836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.543079 - 0.756189i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543079 - 0.756189i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.427i)T \) |
| 7 | \( 1 + (2.31 + 1.28i)T \) |
good | 3 | \( 1 + (0.281 + 1.86i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-2.47 - 0.970i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.464 + 1.50i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.149 + 0.0341i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.505 + 0.344i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-6.85 + 3.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.07 + 1.41i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.12 + 2.33i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.17 - 3.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.622 + 0.0466i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (5.51 + 6.91i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.95 + 3.15i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.42 - 4.10i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.0118 - 0.000887i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (4.81 - 1.89i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-8.84 + 0.662i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-12.0 - 6.98i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 5.21i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.24 - 3.94i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.68 - 6.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.36 + 0.767i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (6.79 - 2.09i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 1.13T + 97T^{2} \) |
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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
−10.91540027337151867224846066559, −9.926012278708567896651062900443, −9.489105225496300798280851434221, −8.225523208781005265365581958278, −7.04878357911796872767622030511, −6.72011617282509522198705343427, −5.71337366834847834671351929191, −3.41042960684339768897315234693, −2.25117890864405391511422525891, −0.891945072688325235458580047366,
1.73965118206663971347280577248, 3.37191237476116960657767178941, 5.13765262347537397617238703466, 5.76699855325122089141528837217, 6.81225563146452743408193234365, 8.113018367479754974861164648142, 9.318326055242291903200734070087, 9.775892745041006733432455724616, 10.02591364746310360599482713053, 11.28913388429054812187855100755