L(s) = 1 | + (−0.856 − 2.63i)3-s + (−0.809 − 0.587i)5-s + (−1.40 + 4.33i)7-s + (−3.78 + 2.75i)9-s + (−3.01 + 1.37i)11-s + (−2.34 + 1.70i)13-s + (−0.856 + 2.63i)15-s + (5.73 + 4.16i)17-s + (−2.20 − 6.79i)19-s + 12.6·21-s − 1.86·23-s + (0.309 + 0.951i)25-s + (3.77 + 2.74i)27-s + (−0.312 + 0.961i)29-s + (−3.56 + 2.59i)31-s + ⋯ |
L(s) = 1 | + (−0.494 − 1.52i)3-s + (−0.361 − 0.262i)5-s + (−0.532 + 1.63i)7-s + (−1.26 + 0.917i)9-s + (−0.910 + 0.413i)11-s + (−0.650 + 0.472i)13-s + (−0.221 + 0.680i)15-s + (1.39 + 1.01i)17-s + (−0.506 − 1.55i)19-s + 2.75·21-s − 0.389·23-s + (0.0618 + 0.190i)25-s + (0.726 + 0.527i)27-s + (−0.0579 + 0.178i)29-s + (−0.640 + 0.465i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0884 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0884 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.200895 + 0.219522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.200895 + 0.219522i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.01 - 1.37i)T \) |
good | 3 | \( 1 + (0.856 + 2.63i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (1.40 - 4.33i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (2.34 - 1.70i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.73 - 4.16i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.20 + 6.79i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + (0.312 - 0.961i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.56 - 2.59i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.66 - 5.12i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.38 + 4.26i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + (-2.42 - 7.46i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.578 - 0.419i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.17 - 3.61i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.71 + 1.24i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + (3.40 + 2.47i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.422 + 1.29i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.90 + 7.19i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-13.2 - 9.60i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-8.78 + 6.38i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.86472412564002676425567520412, −10.65142559794334220450370315870, −9.386313504510489270529791085051, −8.448369208296757331481235268770, −7.63244265566256790718975190460, −6.70483256798284242023985586625, −5.81010879955476168219040685906, −4.99364846835766617383004822763, −2.90264815076706215973267923311, −1.85890763937251369479753381001,
0.18913102105388060178570814365, 3.27174035364000697385512155045, 3.88231334420141329363531397332, 4.98606592731056617773861289300, 5.89280240615496618832537294728, 7.32176358715872517746833744113, 8.013804658218023559381990701839, 9.584058941868415356270702091958, 10.30573600579157682682740671760, 10.42877116340355920903628973677