| L(s) = 1 | + (1.81 − 0.834i)2-s + (2.60 − 3.03i)4-s − 6.14·5-s − 0.590i·7-s + (2.20 − 7.68i)8-s + (−11.1 + 5.12i)10-s − 17.4i·11-s + 1.78·13-s + (−0.492 − 1.07i)14-s + (−2.39 − 15.8i)16-s − 16.9·17-s − 19.5i·19-s + (−16.0 + 18.6i)20-s + (−14.5 − 31.7i)22-s − 7.93i·23-s + ⋯ |
| L(s) = 1 | + (0.908 − 0.417i)2-s + (0.651 − 0.758i)4-s − 1.22·5-s − 0.0843i·7-s + (0.276 − 0.961i)8-s + (−1.11 + 0.512i)10-s − 1.58i·11-s + 0.137·13-s + (−0.0352 − 0.0766i)14-s + (−0.149 − 0.988i)16-s − 0.995·17-s − 1.02i·19-s + (−0.801 + 0.932i)20-s + (−0.662 − 1.44i)22-s − 0.344i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.775848 - 1.69022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.775848 - 1.69022i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.81 + 0.834i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6.14T + 25T^{2} \) |
| 7 | \( 1 + 0.590iT - 49T^{2} \) |
| 11 | \( 1 + 17.4iT - 121T^{2} \) |
| 13 | \( 1 - 1.78T + 169T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 + 19.5iT - 361T^{2} \) |
| 23 | \( 1 + 7.93iT - 529T^{2} \) |
| 29 | \( 1 + 6.35T + 841T^{2} \) |
| 31 | \( 1 - 31.9iT - 961T^{2} \) |
| 37 | \( 1 - 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.33T + 1.68e3T^{2} \) |
| 43 | \( 1 - 39.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 75.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 36.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 37.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 27.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.1T + 9.40e3T^{2} \) |
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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
−11.22694357732818149352743911628, −10.68927065988096871046987572890, −9.160392423062667189795296740328, −8.165495705807101037806057889756, −7.00981403927853196860410831306, −6.03603602576612333745984106099, −4.73747546398372825344894007487, −3.80392860387399370354517765240, −2.77917679961882609003043094855, −0.64591862760490925403838855967,
2.27371969722570352120463837926, 3.90537198580977447128822548668, 4.42664485140482178386009563521, 5.76402015053504925644958479465, 7.05197028698990940083385452721, 7.61107371369650091585527027362, 8.586543390963309434616874085509, 9.972780765405770054220815021228, 11.23315111007215879623493524387, 11.86119744132037729771652354103
