| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.907 + 0.761i)5-s + (−0.173 + 0.984i)6-s + (0.733 − 1.27i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−1.11 + 0.405i)10-s + (−0.592 − 1.02i)11-s + (−0.5 + 0.866i)12-s + (0.446 − 2.53i)13-s + (1.12 − 0.943i)14-s + (−0.907 − 0.761i)15-s + (0.173 + 0.984i)16-s + (−2.09 − 0.761i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.405 + 0.340i)5-s + (−0.0708 + 0.402i)6-s + (0.277 − 0.480i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.352 + 0.128i)10-s + (−0.178 − 0.309i)11-s + (−0.144 + 0.249i)12-s + (0.123 − 0.703i)13-s + (0.300 − 0.252i)14-s + (−0.234 − 0.196i)15-s + (0.0434 + 0.246i)16-s + (−0.507 − 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30866 + 0.568157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30866 + 0.568157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.819 + 4.28i)T \) |
| good | 5 | \( 1 + (0.907 - 0.761i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.733 + 1.27i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.592 + 1.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.446 + 2.53i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.09 + 0.761i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.907 - 0.761i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (8.84 - 3.21i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.96 + 6.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.0641T + 37T^{2} \) |
| 41 | \( 1 + (-1.68 - 9.53i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.55 - 4.65i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.57 + 1.66i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.11 + 0.934i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (1.31 + 0.480i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.97 - 5.01i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.19 - 2.98i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-11.2 + 9.40i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.13 - 6.44i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.24 - 12.7i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.94 - 3.37i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.15 + 12.2i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (12.1 + 4.41i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.70530260465124820503798261303, −12.99274394598256169398033348207, −11.38824830432563180204048280507, −10.96909880938838177427662708753, −9.532694645654465795478412405108, −8.113911996818892250902138331520, −7.09063014282220863996841281884, −5.58106637495611631024911028452, −4.32845814376477998433386883903, −3.05582050226642035351720467508,
2.05466158097569299202966968004, 3.93929794582078748733520256717, 5.35393172554789353890432206348, 6.66287871492417793104828576527, 7.936188165589858494918879833570, 9.072481838681824571911407321462, 10.57437694527063090735112771592, 11.82215812169877422948423745823, 12.30615117646529642548000777059, 13.39958343894648652742468280324
