| L(s) = 1 | + (−1.02 − 0.401i)2-s + (−0.580 − 0.538i)4-s + (−1.00 − 0.687i)5-s + (2.62 − 0.364i)7-s + (1.33 + 2.76i)8-s + (0.755 + 1.10i)10-s + (−0.0890 − 0.590i)11-s + (2.95 − 2.35i)13-s + (−2.82 − 0.679i)14-s + (−0.133 − 1.78i)16-s + (−2.61 − 0.808i)17-s + (0.127 − 0.0734i)19-s + (0.214 + 0.941i)20-s + (−0.146 + 0.640i)22-s + (−2.07 − 6.71i)23-s + ⋯ |
| L(s) = 1 | + (−0.723 − 0.283i)2-s + (−0.290 − 0.269i)4-s + (−0.450 − 0.307i)5-s + (0.990 − 0.137i)7-s + (0.470 + 0.977i)8-s + (0.238 + 0.350i)10-s + (−0.0268 − 0.178i)11-s + (0.819 − 0.653i)13-s + (−0.755 − 0.181i)14-s + (−0.0334 − 0.446i)16-s + (−0.635 − 0.195i)17-s + (0.0292 − 0.0168i)19-s + (0.0480 + 0.210i)20-s + (−0.0311 + 0.136i)22-s + (−0.431 − 1.39i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.412298 - 0.622823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412298 - 0.622823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.364i)T \) |
| good | 2 | \( 1 + (1.02 + 0.401i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (1.00 + 0.687i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.0890 + 0.590i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-2.95 + 2.35i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.61 + 0.808i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.127 + 0.0734i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.07 + 6.71i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (2.75 - 0.628i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (4.54 + 2.62i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.323 - 0.300i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 2.39i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.07 - 0.517i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.29 - 3.30i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-8.63 + 9.30i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.25 + 0.858i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (9.59 + 10.3i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.84 - 1.56i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (14.5 - 5.69i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-5.62 - 9.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.09 + 5.14i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.27 - 0.493i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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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
−10.90065185070907599168262660509, −10.00639474612473120485073622239, −8.822923597816386357216354724704, −8.366401870188092165293974895063, −7.55621499123487557101169051318, −6.02619134257371748230632312655, −4.95029697787852921029160094899, −4.03790022138479593119271032724, −2.13492967080351447511431834771, −0.64786181518213158073077090665,
1.62721877149351068458249528010, 3.61007919565813385666159313605, 4.46999510885093337903064151464, 5.84825924873216788145578835626, 7.23918865828213779080359665864, 7.69597091234727365918760476197, 8.756225726678735442546208293599, 9.263706681878415466787151693854, 10.52307051100175908142258434848, 11.30542091347337477092815073562
