| L(s) = 1 | + (−0.698 − 0.350i)2-s + (0.295 − 1.70i)3-s + (−0.829 − 1.11i)4-s + (0.424 + 1.41i)5-s + (−0.804 + 1.08i)6-s + (−1.50 − 3.49i)7-s + (0.459 + 2.60i)8-s + (−2.82 − 1.00i)9-s + (0.200 − 1.13i)10-s + (4.91 − 1.16i)11-s + (−2.14 + 1.08i)12-s + (3.75 + 2.46i)13-s + (−0.172 + 2.96i)14-s + (2.54 − 0.305i)15-s + (−0.203 + 0.678i)16-s + (−4.30 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.493 − 0.248i)2-s + (0.170 − 0.985i)3-s + (−0.414 − 0.557i)4-s + (0.189 + 0.634i)5-s + (−0.328 + 0.444i)6-s + (−0.569 − 1.31i)7-s + (0.162 + 0.922i)8-s + (−0.941 − 0.336i)9-s + (0.0635 − 0.360i)10-s + (1.48 − 0.351i)11-s + (−0.619 + 0.313i)12-s + (1.04 + 0.684i)13-s + (−0.0461 + 0.792i)14-s + (0.657 − 0.0789i)15-s + (−0.0507 + 0.169i)16-s + (−1.04 + 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0844 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0844 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.496281 - 0.540096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496281 - 0.540096i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.295 + 1.70i)T \) |
| good | 2 | \( 1 + (0.698 + 0.350i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.424 - 1.41i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (1.50 + 3.49i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-4.91 + 1.16i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.75 - 2.46i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (4.30 - 1.56i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.19 - 1.52i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.36 - 3.17i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.0379 + 0.651i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (0.653 - 0.0764i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (-0.766 - 0.642i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (0.610 - 0.306i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (5.75 + 6.10i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (7.88 + 0.921i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (2.07 - 3.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.12 - 1.21i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (4.05 - 5.44i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (0.308 - 5.29i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (1.06 - 6.03i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (0.764 + 4.33i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 5.04i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-1.44 - 0.727i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (0.181 + 1.02i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.43 - 4.77i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−13.80148944045719921771933985914, −13.49879495173819428226383899920, −11.66425597579579282209959142288, −10.82127686966203974549450364808, −9.580309265276553322930751335974, −8.545986947317009806529104632011, −6.95469882794567096760772364256, −6.20716105423918308507922175325, −3.77183149691555731189488387005, −1.36100808130734169910796253745,
3.35952941383070042113607964458, 4.84246320891479655791182688482, 6.38720369305874499358519301729, 8.419302666554150289974348568480, 9.086961628229796765628304337304, 9.633697596199545228333293992697, 11.40424308243623245463113444816, 12.50089246117416384816828229687, 13.54298242572879788777540643851, 14.94434381239569772889098973419
