L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−1.5 + 2.59i)10-s + (0.5 − 0.866i)11-s + 2·13-s + (2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + 3·20-s − 0.999·22-s + (−1.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (0.150 − 0.261i)11-s + 0.554·13-s + (0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + 0.670·20-s − 0.213·22-s + (−0.312 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4616681850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4616681850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−8.907002711422960175695821413280, −8.666728885530200608808099551456, −7.88124647680187511724442287363, −6.75950030300683092856086380864, −5.60471159350409321710190664479, −4.83674380366151270680791359249, −3.86754736033787239974227402154, −2.86772751291315523064126674682, −1.56799511130876613348899266478, −0.22007851049380365941551582820,
1.56796508133297508417498966473, 3.33054616202755432960690540686, 3.86057515678570666065059793922, 5.03707543088538741880961131365, 6.40892400549930428434063361189, 6.64411987099068353693731023810, 7.72155545162421651247777236655, 7.998059111966860772730307215758, 9.182614529256250193980075588850, 10.21405442887395465322949143242