L(s) = 1 | + (−0.866 − 1.5i)2-s + (−1.72 + 0.158i)3-s + (−0.5 + 0.866i)4-s + (−0.358 + 2.20i)5-s + (1.73 + 2.44i)6-s − 1.73·8-s + (2.94 − 0.548i)9-s + (3.62 − 1.37i)10-s + (−2.44 − 1.41i)11-s + (0.724 − 1.57i)12-s + 4·13-s + (0.267 − 3.86i)15-s + (2.49 + 4.33i)16-s + (2.44 + 1.41i)17-s + (−3.37 − 3.94i)18-s + ⋯ |
L(s) = 1 | + (−0.612 − 1.06i)2-s + (−0.995 + 0.0917i)3-s + (−0.250 + 0.433i)4-s + (−0.160 + 0.987i)5-s + (0.707 + 0.999i)6-s − 0.612·8-s + (0.983 − 0.182i)9-s + (1.14 − 0.434i)10-s + (−0.738 − 0.426i)11-s + (0.209 − 0.454i)12-s + 1.10·13-s + (0.0691 − 0.997i)15-s + (0.624 + 1.08i)16-s + (0.594 + 0.342i)17-s + (−0.795 − 0.930i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0190818 + 0.287356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0190818 + 0.287356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.72 - 0.158i)T \) |
| 5 | \( 1 + (0.358 - 2.20i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-2.44 - 1.41i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + (8.48 + 4.89i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + (2.44 - 1.41i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.48 - 4.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.24 + 2.44i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.37144916664067983096958588232, −9.490964096286612736812285623309, −8.346841108646959613776324622290, −7.36709744444686206666642284800, −6.13190192212301205667842665594, −5.77366601816153655355051596720, −4.07180543743416624306869501127, −3.13166526079160935753840658232, −1.80217080287764123918181767626, −0.21897463438967241997858508972,
1.37401140284573142477393359906, 3.59499619455327732309988971279, 5.04259632714305600353730510794, 5.52981807019786811092677509006, 6.49767341169123190846437728276, 7.42440547561918341267882160203, 8.057049398694372611716665276317, 8.988781246750450093113768527151, 9.746160578300436942312949799762, 10.77324526749531586876513428426