L(s) = 1 | + (1.11 − 0.868i)2-s + (−0.795 − 1.53i)3-s + (0.492 − 1.93i)4-s + (−2.41 − 0.646i)5-s + (−2.22 − 1.02i)6-s + (2.82 + 1.62i)7-s + (−1.13 − 2.59i)8-s + (−1.73 + 2.44i)9-s + (−3.25 + 1.37i)10-s + (0.356 + 1.32i)11-s + (−3.37 + 0.783i)12-s + (1.42 − 5.32i)13-s + (4.56 − 0.630i)14-s + (0.924 + 4.22i)15-s + (−3.51 − 1.91i)16-s + 5.37·17-s + ⋯ |
L(s) = 1 | + (0.789 − 0.613i)2-s + (−0.459 − 0.888i)3-s + (0.246 − 0.969i)4-s + (−1.07 − 0.289i)5-s + (−0.907 − 0.419i)6-s + (1.06 + 0.615i)7-s + (−0.400 − 0.916i)8-s + (−0.578 + 0.815i)9-s + (−1.02 + 0.434i)10-s + (0.107 + 0.400i)11-s + (−0.974 + 0.226i)12-s + (0.396 − 1.47i)13-s + (1.22 − 0.168i)14-s + (0.238 + 1.09i)15-s + (−0.878 − 0.477i)16-s + 1.30·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.754443 - 1.10524i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.754443 - 1.10524i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.11 + 0.868i)T \) |
| 3 | \( 1 + (0.795 + 1.53i)T \) |
good | 5 | \( 1 + (2.41 + 0.646i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.82 - 1.62i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.356 - 1.32i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 5.32i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + (-4.71 - 4.71i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.88 - 1.66i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.03 - 0.814i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.621 - 1.07i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.86 - 5.86i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.81 - 1.62i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.61 - 6.03i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 3.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.134 - 0.134i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.21 + 0.592i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (2.29 - 0.615i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (0.0300 - 0.112i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.21iT - 71T^{2} \) |
| 73 | \( 1 + 9.75iT - 73T^{2} \) |
| 79 | \( 1 + (-1.11 + 1.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.74 + 1.54i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 6.12iT - 89T^{2} \) |
| 97 | \( 1 + (2.21 - 3.83i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.29789115018299170447205538282, −12.11650771305013634329045899388, −11.26219392592479382166527561682, −10.12783414386255531550791729473, −8.203789216819802333208512537479, −7.58712430709463146370182971053, −5.79669493866983394063078426963, −5.06611823439960373126894274854, −3.36772923371632772323469318794, −1.40574900883502613608654854743,
3.58711421976688960718084464846, 4.34290549746484289352107541506, 5.47850536492007714634706505681, 6.96674607957217473824024358798, 7.907037398481279490853697393911, 9.110461778978960739517546381852, 10.83044329156474713811286265826, 11.55271141539176289320864702043, 12.03797858355511157925627337445, 13.92550693606676533043134358782