| 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
−11.86119744132037729771652354103, −11.23315111007215879623493524387, −9.972780765405770054220815021228, −8.586543390963309434616874085509, −7.61107371369650091585527027362, −7.05197028698990940083385452721, −5.76402015053504925644958479465, −4.42664485140482178386009563521, −3.90537198580977447128822548668, −2.27371969722570352120463837926,
0.64591862760490925403838855967, 2.77917679961882609003043094855, 3.80392860387399370354517765240, 4.73747546398372825344894007487, 6.03603602576612333745984106099, 7.00981403927853196860410831306, 8.165495705807101037806057889756, 9.160392423062667189795296740328, 10.68927065988096871046987572890, 11.22694357732818149352743911628
