| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.305 + 0.111i)3-s + (0.173 + 0.984i)4-s + (0.749 − 4.25i)5-s + (0.305 + 0.111i)6-s + (−1.19 − 2.07i)7-s + (0.500 − 0.866i)8-s + (−2.21 + 1.86i)9-s + (−3.30 + 2.77i)10-s + (−0.5 + 0.866i)11-s + (−0.162 − 0.281i)12-s + (−4.65 − 1.69i)13-s + (−0.416 + 2.36i)14-s + (0.244 + 1.38i)15-s + (−0.939 + 0.342i)16-s + (3.18 + 2.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.176 + 0.0642i)3-s + (0.0868 + 0.492i)4-s + (0.335 − 1.90i)5-s + (0.124 + 0.0454i)6-s + (−0.452 − 0.784i)7-s + (0.176 − 0.306i)8-s + (−0.738 + 0.620i)9-s + (−1.04 + 0.877i)10-s + (−0.150 + 0.261i)11-s + (−0.0469 − 0.0813i)12-s + (−1.29 − 0.469i)13-s + (−0.111 + 0.630i)14-s + (0.0630 + 0.357i)15-s + (−0.234 + 0.0855i)16-s + (0.773 + 0.648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.000910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000255619 + 0.561302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000255619 + 0.561302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4.07 - 1.55i)T \) |
| good | 3 | \( 1 + (0.305 - 0.111i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.749 + 4.25i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.19 + 2.07i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (4.65 + 1.69i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.18 - 2.67i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.236 + 1.33i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.91 - 5.80i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.51 + 6.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 + (6.40 - 2.33i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.05 + 5.97i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.85 + 6.59i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.959 + 5.44i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.924 - 0.775i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.12 + 6.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.06 - 7.60i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.47 + 8.36i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.03 + 2.19i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.95 + 3.62i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.87 - 3.25i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.1 - 3.69i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.60 + 3.02i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.50376068241909948209612064607, −9.830107471454040335108901716769, −9.093860245209810809770761677899, −8.083281760289242977336463016110, −7.42962600635862700857377675926, −5.61726762665143408183366597116, −5.00810779676662717252223985646, −3.69023707371841238127055832961, −1.93683324647727516866003596415, −0.41530407911414079608680643256,
2.48010979242215319377023475584, 3.25085564142871810956987332145, 5.45751099048262082893931754433, 6.10567078338172515431034220186, 7.06402822471437515630792779790, 7.64690117546261197534381045084, 9.311953870486468131521860344084, 9.595438784243757444179689613888, 10.68262523625992173744000130015, 11.54272850902478874986024605358
