| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (1.72 + 2.98i)5-s + (−0.5 + 0.866i)6-s + (−0.784 − 0.871i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−3.14 − 1.40i)10-s + (0.704 − 0.149i)11-s + (−0.104 − 0.994i)12-s + (−0.287 + 2.73i)13-s + (1.14 + 0.243i)14-s + (2.78 + 2.02i)15-s + (−0.809 − 0.587i)16-s + (4.31 + 0.916i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (0.770 + 1.33i)5-s + (−0.204 + 0.353i)6-s + (−0.296 − 0.329i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.995 − 0.443i)10-s + (0.212 − 0.0451i)11-s + (−0.0301 − 0.287i)12-s + (−0.0796 + 0.757i)13-s + (0.306 + 0.0651i)14-s + (0.719 + 0.522i)15-s + (−0.202 − 0.146i)16-s + (1.04 + 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06052 + 0.461387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06052 + 0.461387i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 31 | \( 1 + (5.51 - 0.782i)T \) |
| good | 5 | \( 1 + (-1.72 - 2.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.784 + 0.871i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-0.704 + 0.149i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.287 - 2.73i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-4.31 - 0.916i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.349 + 3.32i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.193 - 0.596i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.67i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-5.55 + 9.62i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.38 + 4.18i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (1.04 + 9.95i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (4.16 + 3.02i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.83 - 7.59i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-9.18 + 4.08i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 3.58T + 61T^{2} \) |
| 67 | \( 1 + (-5.86 - 10.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.37 - 3.74i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (0.469 - 0.0997i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (1.41 + 0.301i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (13.0 + 5.79i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (3.14 - 9.68i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.755 - 2.32i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.96511912239351227954412937225, −11.49020496793699163885294220088, −10.47680274791817432758306146013, −9.726062603978746783490763905943, −8.797593267306523033127942510858, −7.30499876875215175451920254279, −6.85038358267488483887699919165, −5.66801374666825783957561188418, −3.53915636415017717528503706540, −2.07186961657817909635797849217,
1.50850930842737722745456573496, 3.19229982523924610257675734386, 4.83711676568749471235692940153, 6.02658038933944388996574752812, 7.84319648602655804983830193783, 8.576781752558840650880483322299, 9.676820050600201323416707792992, 9.936446745930170247514977760954, 11.52627305921822939463242554142, 12.67598444462184457278435817253
