| L(s) = 1 | + (0.768 + 0.110i)3-s + (−2.23 − 0.0976i)5-s + (−2.41 + 2.09i)7-s + (−2.29 − 0.675i)9-s + (3.09 + 1.99i)11-s + (−1.64 − 1.42i)13-s + (−1.70 − 0.321i)15-s + (−6.67 + 3.04i)17-s + (−1.21 + 2.66i)19-s + (−2.08 + 1.34i)21-s + (−4.38 − 1.95i)23-s + (4.98 + 0.436i)25-s + (−3.81 − 1.74i)27-s + (2.38 + 5.22i)29-s + (−0.218 − 1.52i)31-s + ⋯ |
| L(s) = 1 | + (0.443 + 0.0638i)3-s + (−0.999 − 0.0436i)5-s + (−0.911 + 0.790i)7-s + (−0.766 − 0.225i)9-s + (0.933 + 0.600i)11-s + (−0.457 − 0.395i)13-s + (−0.440 − 0.0831i)15-s + (−1.61 + 0.739i)17-s + (−0.279 + 0.611i)19-s + (−0.455 + 0.292i)21-s + (−0.913 − 0.407i)23-s + (0.996 + 0.0872i)25-s + (−0.733 − 0.335i)27-s + (0.442 + 0.969i)29-s + (−0.0392 − 0.273i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.109855 + 0.427147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.109855 + 0.427147i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.0976i)T \) |
| 23 | \( 1 + (4.38 + 1.95i)T \) |
| good | 3 | \( 1 + (-0.768 - 0.110i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (2.41 - 2.09i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.09 - 1.99i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.64 + 1.42i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.67 - 3.04i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.21 - 2.66i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.38 - 5.22i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.218 + 1.52i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.256 - 0.873i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.53 + 1.62i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 0.570i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 4.46iT - 47T^{2} \) |
| 53 | \( 1 + (7.82 - 6.78i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.621 - 0.717i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.46 + 10.2i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.77 - 5.87i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (8.76 - 5.63i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-8.40 - 3.83i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.32 + 3.84i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.28 + 14.6i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.36 - 9.47i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.287 - 0.979i)T + (-81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55904736366563016008921194960, −10.56399683903897063080421355288, −9.368540813233729648310441650146, −8.797611669239691197457671027007, −7.984538312668020740818627101710, −6.75515850946494618234187222443, −5.99350387334714975616199635788, −4.44928804761273126667724580874, −3.52570624509664902366004559759, −2.36709221616171126349312322197,
0.24190166030041850595460170074, 2.61876179408216383584933481291, 3.72558815575553422852395167021, 4.55101362760449232170670672033, 6.26516894226639341613101602322, 6.99643841834279162659459267164, 7.956668975646960865576575272518, 8.913099610073768162648400759349, 9.564343206247868269667063816778, 10.94606096354340363137575939401
