| L(s) = 1 | + (1.26 − 1.54i)2-s + (−2.40 + 1.79i)3-s + (−0.801 − 3.91i)4-s + (−7.02 + 2.55i)5-s + (−0.265 + 5.99i)6-s + (−8.30 + 1.46i)7-s + (−7.08 − 3.71i)8-s + (2.57 − 8.62i)9-s + (−4.92 + 14.1i)10-s + (3.00 − 8.26i)11-s + (8.95 + 7.99i)12-s + (5.39 + 4.53i)13-s + (−8.23 + 14.7i)14-s + (12.3 − 18.7i)15-s + (−14.7 + 6.27i)16-s + (−10.7 + 18.6i)17-s + ⋯ |
| L(s) = 1 | + (0.632 − 0.774i)2-s + (−0.801 + 0.597i)3-s + (−0.200 − 0.979i)4-s + (−1.40 + 0.511i)5-s + (−0.0442 + 0.999i)6-s + (−1.18 + 0.209i)7-s + (−0.885 − 0.464i)8-s + (0.286 − 0.958i)9-s + (−0.492 + 1.41i)10-s + (0.273 − 0.751i)11-s + (0.745 + 0.666i)12-s + (0.415 + 0.348i)13-s + (−0.588 + 1.05i)14-s + (0.821 − 1.25i)15-s + (−0.919 + 0.392i)16-s + (−0.634 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0188523 + 0.0848318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0188523 + 0.0848318i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 1.54i)T \) |
| 3 | \( 1 + (2.40 - 1.79i)T \) |
| good | 5 | \( 1 + (7.02 - 2.55i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (8.30 - 1.46i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-3.00 + 8.26i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-5.39 - 4.53i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (10.7 - 18.6i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-22.4 + 12.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (40.0 + 7.05i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (6.59 - 5.53i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (11.2 + 1.98i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-6.02 + 10.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (57.3 + 48.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-5.67 + 15.5i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 1.80i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 6.32T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.9 - 46.4i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-0.604 - 3.42i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (40.3 - 48.1i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-8.73 - 5.04i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (13.0 + 22.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-61.6 - 73.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-2.03 - 2.41i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (23.2 + 40.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.79 + 3.20i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.50080795155661504141468643374, −11.73728951445678079438304388967, −11.03604780971570122657049965379, −10.07579720172111169352387671619, −8.842433411633840550630971557088, −6.79098550735262582733755035599, −5.82078983670125491292191292963, −4.06510866342935438948849581608, −3.41526863098032366066301041105, −0.05590765408212433639575420227,
3.60319875623369859146575996813, 4.85560650714999189440274411926, 6.26325437563436235681294692655, 7.29503891481586540216353127333, 8.058663072319532173923890632667, 9.702749812840019697501813157248, 11.58032302887480542084879598135, 12.08031325643230771716079950306, 12.94670502309032175139753445478, 13.81800895103785296402768558707
