L(s) = 1 | + (1.18 − 2.58i)3-s + (0.654 − 0.755i)5-s + (−1.60 + 1.02i)7-s + (−3.32 − 3.84i)9-s + (0.455 − 3.16i)11-s + (−3.20 − 2.05i)13-s + (−1.18 − 2.58i)15-s + (3.02 + 0.888i)17-s + (0.203 − 0.0596i)19-s + (0.770 + 5.35i)21-s + (4.68 − 1.03i)23-s + (−0.142 − 0.989i)25-s + (−5.68 + 1.66i)27-s + (−4.81 − 1.41i)29-s + (2.57 + 5.62i)31-s + ⋯ |
L(s) = 1 | + (0.681 − 1.49i)3-s + (0.292 − 0.337i)5-s + (−0.605 + 0.389i)7-s + (−1.10 − 1.28i)9-s + (0.137 − 0.954i)11-s + (−0.888 − 0.571i)13-s + (−0.304 − 0.667i)15-s + (0.733 + 0.215i)17-s + (0.0465 − 0.0136i)19-s + (0.168 + 1.16i)21-s + (0.976 − 0.215i)23-s + (−0.0284 − 0.197i)25-s + (−1.09 + 0.321i)27-s + (−0.893 − 0.262i)29-s + (0.461 + 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.795060 - 1.39305i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.795060 - 1.39305i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.68 + 1.03i)T \) |
good | 3 | \( 1 + (-1.18 + 2.58i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (1.60 - 1.02i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.455 + 3.16i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.20 + 2.05i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.02 - 0.888i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.203 + 0.0596i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (4.81 + 1.41i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.57 - 5.62i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.32 - 1.52i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (4.97 - 5.74i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.18 + 9.16i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + (2.46 - 1.58i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.02 - 1.94i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.657 - 1.44i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (2.15 + 14.9i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.82 - 12.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 3.41i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.25 - 4.66i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.94 - 4.55i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.324 + 0.710i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.73 + 4.31i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.80869598592060760084246609232, −9.593634208858823895794140101573, −8.789331007426238484835524165956, −8.016758326955328050724870618686, −7.13766387377789597896025426092, −6.22058156653596608022092287920, −5.31217747940249945150946384128, −3.34145109627180917202935822711, −2.45558200780583941694976248232, −0.954566678458559634755111496035,
2.42291906138981236345515070294, 3.52822559793342673827787586392, 4.44064688000869807613530482550, 5.40491822079768308347730515362, 6.86194766478440262126167278226, 7.72583034955636752416674357223, 9.167743659442799921406168786181, 9.576690024320334993380537270380, 10.17665793610518834194812560838, 11.03243477883028824493286419609