| 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
−13.39958343894648652742468280324, −12.30615117646529642548000777059, −11.82215812169877422948423745823, −10.57437694527063090735112771592, −9.072481838681824571911407321462, −7.936188165589858494918879833570, −6.66287871492417793104828576527, −5.35393172554789353890432206348, −3.93929794582078748733520256717, −2.05466158097569299202966968004,
3.05582050226642035351720467508, 4.32845814376477998433386883903, 5.58106637495611631024911028452, 7.09063014282220863996841281884, 8.113911996818892250902138331520, 9.532694645654465795478412405108, 10.96909880938838177427662708753, 11.38824830432563180204048280507, 12.99274394598256169398033348207, 13.70530260465124820503798261303
