L(s) = 1 | + (0.300 + 1.70i)2-s + (−0.939 + 0.342i)4-s + (−1.32 − 1.11i)5-s + (−1.87 − 0.684i)7-s + (0.866 + 1.50i)8-s + (1.49 − 2.59i)10-s + (−2.65 + 2.22i)11-s + (−0.173 + 0.984i)13-s + (0.601 − 3.41i)14-s + (−3.83 + 3.21i)16-s + (−2.59 + 4.5i)17-s + (−1 − 1.73i)19-s + (1.62 + 0.592i)20-s + (−4.59 − 3.85i)22-s + (−3.25 + 1.18i)23-s + ⋯ |
L(s) = 1 | + (0.212 + 1.20i)2-s + (−0.469 + 0.171i)4-s + (−0.593 − 0.497i)5-s + (−0.710 − 0.258i)7-s + (0.306 + 0.530i)8-s + (0.474 − 0.821i)10-s + (−0.800 + 0.671i)11-s + (−0.0481 + 0.273i)13-s + (0.160 − 0.911i)14-s + (−0.957 + 0.803i)16-s + (−0.630 + 1.09i)17-s + (−0.229 − 0.397i)19-s + (0.363 + 0.132i)20-s + (−0.979 − 0.822i)22-s + (−0.678 + 0.247i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.167184 - 0.558435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.167184 - 0.558435i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.300 - 1.70i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (1.32 + 1.11i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.87 + 0.684i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.65 - 2.22i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.59 - 4.5i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.25 - 1.18i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.300 - 1.70i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (7.51 - 2.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.20 - 6.82i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.28i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (6.51 + 2.36i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-10.6 - 8.90i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-6.57 - 2.39i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.73 - 9.84i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.19 + 9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.347 - 1.96i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.40 + 13.6i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.53 + 1.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.82585971676707798120650303427, −9.986011797223396158525297234519, −8.841213903909955588338328810937, −8.139015259228360777288296151511, −7.31889577915189228500937524702, −6.61613345741667317973520037462, −5.68007140144769493723470332359, −4.69693524470773182655804537694, −3.86388735675631636893864464163, −2.14280854088380972807815994625,
0.25732361612947361116452664767, 2.25357579427254747573971788132, 3.12923648066910479015380413594, 3.85767450466375343022856237533, 5.13968686910292797569788124667, 6.36126391357485455946790722139, 7.29277956039818256551040306249, 8.190677185793186205107738896923, 9.418404108676126764526282557792, 10.04293983383212639614970609897