| L(s) = 1 | + (0.260 − 0.800i)2-s + (−1.08 + 1.20i)3-s + (1.04 + 0.758i)4-s + (0.682 + 1.18i)6-s + (−0.454 + 4.32i)7-s + (2.24 − 1.62i)8-s + (0.0392 + 0.373i)9-s + (−3.51 + 1.56i)11-s + (−2.04 + 0.434i)12-s + (−0.0941 − 0.0200i)13-s + (3.34 + 1.48i)14-s + (0.0765 + 0.235i)16-s + (−6.48 − 2.88i)17-s + (0.309 + 0.0657i)18-s + (−2.31 + 0.491i)19-s + ⋯ |
| L(s) = 1 | + (0.184 − 0.566i)2-s + (−0.625 + 0.695i)3-s + (0.522 + 0.379i)4-s + (0.278 + 0.482i)6-s + (−0.171 + 1.63i)7-s + (0.792 − 0.575i)8-s + (0.0130 + 0.124i)9-s + (−1.05 + 0.471i)11-s + (−0.590 + 0.125i)12-s + (−0.0261 − 0.00554i)13-s + (0.893 + 0.398i)14-s + (0.0191 + 0.0588i)16-s + (−1.57 − 0.700i)17-s + (0.0728 + 0.0154i)18-s + (−0.530 + 0.112i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.523098 + 0.982403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.523098 + 0.982403i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (5.46 - 1.07i)T \) |
| good | 2 | \( 1 + (-0.260 + 0.800i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.08 - 1.20i)T + (-0.313 - 2.98i)T^{2} \) |
| 7 | \( 1 + (0.454 - 4.32i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (3.51 - 1.56i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (0.0941 + 0.0200i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (6.48 + 2.88i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (2.31 - 0.491i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-6.43 + 4.67i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 3.33i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.86 - 4.95i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.85 - 2.05i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.04 - 0.434i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-3.40 - 10.4i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.127 - 1.21i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (3.30 - 3.67i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 + (4.79 - 8.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.971 - 9.24i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.176 - 0.0786i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 5.24i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.841 - 0.934i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (0.445 + 0.323i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (8.36 + 6.07i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.89379253923937391700114181043, −9.967211662988688110298465724135, −9.053312889660626856423418913270, −8.157397243394547056168797707569, −7.02910137599724539497124188124, −6.06810954621124137666492216472, −5.03189239114309415524096769897, −4.40526063075437588024891735684, −2.72863494400024269440774957522, −2.31044623465046127992128916470,
0.52924047173811532988731710071, 1.91092249512717515163995297881, 3.60719026681445398136290757308, 4.83995621523526146973172571570, 5.78378931734976967859972585510, 6.71250347142211552793628219386, 7.10760332802862519968605784706, 7.85109466329218554208761401836, 9.097202095254172737111303826678, 10.42976904707480966059348424785
