| L(s) = 1 | + (−1.59 + 0.821i)2-s + (0.786 − 0.618i)3-s + (0.704 − 0.989i)4-s + (−2.75 + 2.62i)5-s + (−0.744 + 1.63i)6-s + (1.28 − 2.31i)7-s + (0.200 − 1.39i)8-s + (0.235 − 0.971i)9-s + (2.23 − 6.45i)10-s + (2.74 + 1.41i)11-s + (−0.0577 − 1.21i)12-s + (3.50 + 4.05i)13-s + (−0.140 + 4.74i)14-s + (−0.542 + 3.77i)15-s + (1.62 + 4.68i)16-s + (−4.57 + 0.436i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 + 0.580i)2-s + (0.453 − 0.356i)3-s + (0.352 − 0.494i)4-s + (−1.23 + 1.17i)5-s + (−0.304 + 0.665i)6-s + (0.484 − 0.874i)7-s + (0.0708 − 0.492i)8-s + (0.0785 − 0.323i)9-s + (0.706 − 2.04i)10-s + (0.826 + 0.426i)11-s + (−0.0166 − 0.350i)12-s + (0.973 + 1.12i)13-s + (−0.0375 + 1.26i)14-s + (−0.140 + 0.973i)15-s + (0.405 + 1.17i)16-s + (−1.10 + 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.235021 + 0.535119i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235021 + 0.535119i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.786 + 0.618i)T \) |
| 7 | \( 1 + (-1.28 + 2.31i)T \) |
| 23 | \( 1 + (-0.0120 - 4.79i)T \) |
| good | 2 | \( 1 + (1.59 - 0.821i)T + (1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (2.75 - 2.62i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (-2.74 - 1.41i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-3.50 - 4.05i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.57 - 0.436i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (2.44 + 0.233i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (2.52 - 5.53i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (7.11 - 2.84i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (1.38 - 5.69i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (1.60 + 0.471i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.888 - 6.17i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.32 - 1.02i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (3.12 - 9.04i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (4.22 + 3.31i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (0.774 - 16.2i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-8.02 - 5.15i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.11 + 7.18i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-4.19 + 0.808i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (4.43 - 1.30i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.94 - 1.17i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (1.13 + 0.333i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.10996761741343216600446259226, −10.44316069896998795787619224790, −9.156754835876638979792893776689, −8.560176782962551352859622975289, −7.59346840966453341858636037095, −6.94718189131573287036576864759, −6.62457427894686107354759491543, −4.13174341380071875789149029406, −3.66089251091153106185675135741, −1.57163635660797969281098818926,
0.51856296128093503076758306948, 2.07167409327109307762959874954, 3.66911927708852852916037501337, 4.68878440224565289323449810317, 5.83632598844260723101339502941, 7.69624064785707850182141500209, 8.413468159194954188483183191678, 8.801640327594608959987139630337, 9.371446946418044597148271257984, 10.91960768755541495242576773905
