| L(s) = 1 | + (−1.01 − 1.21i)2-s + (−0.592 + 1.62i)3-s + (0.260 − 1.47i)4-s + (−0.649 − 3.68i)5-s + (2.57 − 0.937i)6-s + (6.08 − 10.5i)7-s + (−7.53 + 4.34i)8-s + (−2.29 − 1.92i)9-s + (−3.80 + 4.53i)10-s + (−1.24 − 2.16i)11-s + (2.24 + 1.29i)12-s + (3.75 + 10.3i)13-s + (−18.9 + 3.34i)14-s + (6.37 + 1.12i)15-s + (7.29 + 2.65i)16-s + (9.80 − 8.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.508 − 0.605i)2-s + (−0.197 + 0.542i)3-s + (0.0650 − 0.368i)4-s + (−0.129 − 0.736i)5-s + (0.429 − 0.156i)6-s + (0.869 − 1.50i)7-s + (−0.941 + 0.543i)8-s + (−0.255 − 0.214i)9-s + (−0.380 + 0.453i)10-s + (−0.113 − 0.196i)11-s + (0.187 + 0.108i)12-s + (0.289 + 0.794i)13-s + (−1.35 + 0.238i)14-s + (0.425 + 0.0749i)15-s + (0.456 + 0.165i)16-s + (0.576 − 0.483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.586721 - 0.643324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586721 - 0.643324i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.592 - 1.62i)T \) |
| 19 | \( 1 + (-7.95 - 17.2i)T \) |
| good | 2 | \( 1 + (1.01 + 1.21i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (0.649 + 3.68i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.08 + 10.5i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (1.24 + 2.16i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.75 - 10.3i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-9.80 + 8.22i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (5.56 - 31.5i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-12.6 + 15.0i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (36.5 + 21.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 9.59iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-22.3 + 61.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-5.64 - 32.0i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-53.8 - 45.1i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-44.6 - 7.86i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-11.8 - 14.1i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (10.5 - 59.7i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-21.1 + 25.2i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (79.2 - 13.9i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (23.7 + 8.62i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (28.5 - 78.4i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-45.4 + 78.7i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (23.0 + 63.3i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-17.2 - 20.5i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−14.54310260526630033114132696829, −13.74423757394524042956216078988, −11.93972562229741293413439923353, −11.07813099832274794377783907145, −10.13866540970901183727195682919, −9.064023392669112510730327081410, −7.61618986744269343437484731022, −5.53695948048854752147410694381, −4.08512988930132767765395440755, −1.16191882997487178794272031566,
2.77690548510357746992755706306, 5.53358634072438869774866771401, 6.85915715546423948544930458735, 8.048176393418285544711625580737, 8.896348336272586116455284856813, 10.78881044602390379398008422409, 11.98690469119027281372382439302, 12.76006378468710321538909522869, 14.58644362503640591225307618645, 15.25244200775408465810854309048
