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 | \( 1 - 2.22iT - 97T^{2} \) |
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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
−14.45314074891695813781704254318, −13.02670378472094650758571005493, −11.87788132747236683875934005342, −10.86773761939265056640030476802, −9.950423335140631752223800189162, −8.665614446311123809430241695195, −8.046532670337603173301211466347, −6.13455913690211485644224156598, −4.13259039340064034920323624195, −2.32893792479702505689800989009,
1.77756495730142384033954453433, 4.67762246919378936523647647867, 6.53513172287902680440732507763, 7.35753520429612329507760643649, 8.625627145943266633493381068039, 9.515565619828406398793387636790, 10.94594719852831285667298096024, 11.85012995652094919708135537700, 13.65202769915119487663106068878, 14.20771852109056533605840313278