| L(s) = 1 | + (−0.212 + 0.977i)2-s + (−0.800 + 0.599i)3-s + (−0.909 − 0.415i)4-s + (2.00 − 0.993i)5-s + (−0.415 − 0.909i)6-s + (0.0316 − 0.441i)7-s + (0.599 − 0.800i)8-s + (0.281 − 0.959i)9-s + (0.544 + 2.16i)10-s + (−0.135 + 0.210i)11-s + (0.977 − 0.212i)12-s + (−0.284 + 0.0203i)13-s + (0.425 + 0.124i)14-s + (−1.00 + 1.99i)15-s + (0.654 + 0.755i)16-s + (0.163 − 0.438i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (−0.462 + 0.345i)3-s + (−0.454 − 0.207i)4-s + (0.895 − 0.444i)5-s + (−0.169 − 0.371i)6-s + (0.0119 − 0.167i)7-s + (0.211 − 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.172 + 0.685i)10-s + (−0.0407 + 0.0634i)11-s + (0.282 − 0.0613i)12-s + (−0.0788 + 0.00563i)13-s + (0.113 + 0.0333i)14-s + (−0.260 + 0.515i)15-s + (0.163 + 0.188i)16-s + (0.0396 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31302 + 0.208841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31302 + 0.208841i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 3 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (-2.00 + 0.993i)T \) |
| 23 | \( 1 + (-1.08 + 4.67i)T \) |
| good | 7 | \( 1 + (-0.0316 + 0.441i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.135 - 0.210i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.284 - 0.0203i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-0.163 + 0.438i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 5.09i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-9.35 + 4.27i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.295 - 2.05i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.51 - 6.44i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (2.15 - 0.631i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.34 - 4.46i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (7.13 - 7.13i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.51 + 0.179i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (6.72 + 5.82i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-3.43 + 0.493i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.30 - 0.719i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (3.68 - 2.36i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 4.64i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-3.67 + 4.23i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (6.44 - 3.51i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-1.64 + 11.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.83 - 0.999i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.25731568519680740885077914049, −9.642591141935633668304608584559, −8.859699129392155778917638767822, −7.955125922530806604451730919732, −6.69187550380564470130645843176, −6.19502377585682497947278665752, −4.98539539156653526576376538257, −4.57770779591718315928296551654, −2.78005767940532945976533055383, −0.936619522343447224716488734241,
1.31651210354496584265461242529, 2.45835864746992257991436422401, 3.63522910189428979499035491655, 5.11759355347409078460757201656, 5.84280548870109271127810896941, 6.85308476316322050927189068222, 7.83850829442417316236129377471, 8.904756545964622313787085351171, 9.819765683695208393029102597191, 10.41249702390132463922873600167
