| L(s) = 1 | + (−0.257 + 2.45i)2-s + (1.08 + 1.20i)3-s + (−3.99 − 0.849i)4-s + (3.16 + 1.41i)5-s + (−3.22 + 2.34i)6-s + (1.58 − 4.89i)8-s + (0.0399 − 0.379i)9-s + (−4.27 + 7.40i)10-s + (−0.0978 + 3.31i)11-s + (−3.30 − 5.72i)12-s + (−0.528 − 0.384i)13-s + (1.73 + 5.33i)15-s + (4.12 + 1.83i)16-s + (−0.118 − 1.13i)17-s + (0.921 + 0.195i)18-s + (−5.94 + 1.26i)19-s + ⋯ |
| L(s) = 1 | + (−0.182 + 1.73i)2-s + (0.625 + 0.694i)3-s + (−1.99 − 0.424i)4-s + (1.41 + 0.630i)5-s + (−1.31 + 0.957i)6-s + (0.561 − 1.72i)8-s + (0.0133 − 0.126i)9-s + (−1.35 + 2.34i)10-s + (−0.0295 + 0.999i)11-s + (−0.953 − 1.65i)12-s + (−0.146 − 0.106i)13-s + (0.447 + 1.37i)15-s + (1.03 + 0.459i)16-s + (−0.0288 − 0.274i)17-s + (0.217 + 0.0461i)18-s + (−1.36 + 0.289i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228376 - 1.71774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228376 - 1.71774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.0978 - 3.31i)T \) |
| good | 2 | \( 1 + (0.257 - 2.45i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-1.08 - 1.20i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-3.16 - 1.41i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (0.528 + 0.384i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.118 + 1.13i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (5.94 - 1.26i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.33 - 5.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.41 + 4.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.55 - 1.13i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.294 - 0.326i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.82 + 5.61i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (0.591 - 0.125i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-8.97 + 3.99i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.65 - 0.352i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (6.26 + 2.78i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-3.08 + 5.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.38 + 3.18i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.55 - 1.39i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.277 - 2.63i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.41 + 3.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.349 + 0.604i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.0 + 8.73i)T + (29.9 + 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78830184273439831206505009824, −9.832594367528264705405949556230, −9.454687594158301503527940544093, −8.719083443127999051558225649684, −7.54143526471212141577565383535, −6.75087362675184203662465921629, −5.96510988807423145693896760277, −5.06654123122937442020641280169, −3.94184724637354549165955446889, −2.30879919967879086078265191352,
1.08303561288070800139855714598, 2.13764635942259892076577912316, 2.79339141225435190876328042872, 4.30807993040237172011844188802, 5.44223560730353980237355959274, 6.65495376045035303800330343394, 8.270632539978736992771157693130, 8.842288415890886006590736514355, 9.436878407071260758838205393773, 10.64438876949821741334606609487
