L(s) = 1 | + (−1.18 + 0.773i)2-s + (0.5 − 0.866i)3-s + (0.803 − 1.83i)4-s + (0.380 − 0.219i)5-s + (0.0777 + 1.41i)6-s + (2.02 − 1.70i)7-s + (0.464 + 2.79i)8-s + (−0.499 − 0.866i)9-s + (−0.280 + 0.553i)10-s + (1.83 + 1.05i)11-s + (−1.18 − 1.61i)12-s + 3.84i·13-s + (−1.07 + 3.58i)14-s − 0.438i·15-s + (−2.70 − 2.94i)16-s + (−4.89 − 2.82i)17-s + ⋯ |
L(s) = 1 | + (−0.837 + 0.546i)2-s + (0.288 − 0.499i)3-s + (0.401 − 0.915i)4-s + (0.170 − 0.0981i)5-s + (0.0317 + 0.576i)6-s + (0.764 − 0.644i)7-s + (0.164 + 0.986i)8-s + (−0.166 − 0.288i)9-s + (−0.0886 + 0.175i)10-s + (0.552 + 0.318i)11-s + (−0.341 − 0.465i)12-s + 1.06i·13-s + (−0.288 + 0.957i)14-s − 0.113i·15-s + (−0.676 − 0.736i)16-s + (−1.18 − 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779934 - 0.0431115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779934 - 0.0431115i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.18 - 0.773i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.02 + 1.70i)T \) |
good | 5 | \( 1 + (-0.380 + 0.219i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.83 - 1.05i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.84iT - 13T^{2} \) |
| 17 | \( 1 + (4.89 + 2.82i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.48 + 2.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.13 - 2.38i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 + (3.71 - 6.43i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.64 - 4.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.81iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 + (0.844 + 1.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.35 - 9.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.05 + 7.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.35 + 3.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.79 + 3.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.16iT - 71T^{2} \) |
| 73 | \( 1 + (8.69 + 5.01i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.4 + 7.79i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 + (9.02 - 5.20i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.22iT - 97T^{2} \) |
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
−14.20771852109056533605840313278, −13.65202769915119487663106068878, −11.85012995652094919708135537700, −10.94594719852831285667298096024, −9.515565619828406398793387636790, −8.625627145943266633493381068039, −7.35753520429612329507760643649, −6.53513172287902680440732507763, −4.67762246919378936523647647867, −1.77756495730142384033954453433,
2.32893792479702505689800989009, 4.13259039340064034920323624195, 6.13455913690211485644224156598, 8.046532670337603173301211466347, 8.665614446311123809430241695195, 9.950423335140631752223800189162, 10.86773761939265056640030476802, 11.87788132747236683875934005342, 13.02670378472094650758571005493, 14.45314074891695813781704254318