| L(s) = 1 | + (−0.0752 + 0.0483i)3-s + (0.415 + 0.909i)5-s + (0.0649 − 0.0190i)7-s + (−1.24 + 2.72i)9-s + (1.34 + 1.55i)11-s + (0.710 + 0.208i)13-s + (−0.0752 − 0.0483i)15-s + (−0.424 + 2.95i)17-s + (0.239 + 1.66i)19-s + (−0.00396 + 0.00457i)21-s + (4.73 − 0.742i)23-s + (−0.654 + 0.755i)25-s + (−0.0762 − 0.530i)27-s + (−0.643 + 4.47i)29-s + (1.37 + 0.880i)31-s + ⋯ |
| L(s) = 1 | + (−0.0434 + 0.0279i)3-s + (0.185 + 0.406i)5-s + (0.0245 − 0.00720i)7-s + (−0.414 + 0.907i)9-s + (0.405 + 0.467i)11-s + (0.197 + 0.0578i)13-s + (−0.0194 − 0.0124i)15-s + (−0.102 + 0.715i)17-s + (0.0548 + 0.381i)19-s + (−0.000864 + 0.000997i)21-s + (0.987 − 0.154i)23-s + (−0.130 + 0.151i)25-s + (−0.0146 − 0.102i)27-s + (−0.119 + 0.831i)29-s + (0.246 + 0.158i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13478 + 0.729908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13478 + 0.729908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.73 + 0.742i)T \) |
| good | 3 | \( 1 + (0.0752 - 0.0483i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-0.0649 + 0.0190i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.34 - 1.55i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.710 - 0.208i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.424 - 2.95i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.239 - 1.66i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.643 - 4.47i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 0.880i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.0152 - 0.0333i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (0.321 + 0.704i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.61 + 1.68i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 5.04T + 47T^{2} \) |
| 53 | \( 1 + (-2.31 + 0.679i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (0.859 + 0.252i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.74 + 2.40i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.55 + 2.95i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.08 + 7.02i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.18 + 8.27i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (5.78 + 1.69i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.32 + 11.6i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (6.64 - 4.26i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.82 + 6.17i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.97922083037348266297407238721, −10.55301771702832255974698887602, −9.436740516527015129501532093072, −8.517163977165739794720514062276, −7.54736868127591842309354154950, −6.59140988704052768490979488531, −5.56563618186141561332939687100, −4.49499379633427567558646040707, −3.15635081404297026095433683106, −1.80784469481552528637251225500,
0.900391333522137424233631708178, 2.78036945109138412758297939525, 3.99384140794056146875225945430, 5.24184477141102707663566589856, 6.19026164697130118373162144052, 7.10783843534039294526889911783, 8.365828439999277675433130818112, 9.118319983970737042766052247548, 9.795709073428376275802938451373, 11.11929327809351257488939770321
