L(s) = 1 | + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.0747 + 0.997i)5-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)10-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)26-s + ⋯ |
L(s) = 1 | + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.0747 + 0.997i)5-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)10-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1746816274 + 1.033675420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1746816274 + 1.033675420i\) |
\(L(1)\) |
\(\approx\) |
\(0.6469327873 + 0.7892277685i\) |
\(L(1)\) |
\(\approx\) |
\(0.6469327873 + 0.7892277685i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.365 + 0.930i)T \) |
| 5 | \( 1 + (0.0747 + 0.997i)T \) |
| 11 | \( 1 + (0.365 + 0.930i)T \) |
| 13 | \( 1 + (-0.988 - 0.149i)T \) |
| 17 | \( 1 + (-0.222 + 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.733 + 0.680i)T \) |
| 29 | \( 1 + (-0.733 - 0.680i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (0.0747 + 0.997i)T \) |
| 43 | \( 1 + (0.0747 - 0.997i)T \) |
| 47 | \( 1 + (0.365 + 0.930i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.826 + 0.563i)T \) |
| 61 | \( 1 + (-0.733 - 0.680i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.988 + 0.149i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.59812851180920152094318151024, −22.42681282856406370605953587987, −21.87784170760376334836514801303, −20.95845468564544666933679768586, −20.13692107462709363348341495740, −19.62984726304488061336921286208, −18.55331063996036425990639689936, −17.70570386510855153748795449578, −16.61133971045696239159117903378, −15.85978545989790163419480348396, −14.4104174230259993653886452653, −13.890393292393691191235216593200, −12.86738721082742133615784945845, −12.09130616653724407823508261360, −11.41469396041919088869231821379, −10.247138726268978995877411334125, −9.262820503382432602824047450506, −8.73090693762941341508725189072, −7.333434069495798061566187507915, −5.786908517682648705152559594090, −5.08006079694338624635519419224, −4.106899739773066911446711857649, −2.975166476944110360766656861869, −1.75687275722689466011271447268, −0.516458658469297684226994088174,
2.09262158218201882934393047399, 3.37250389154607994679385152155, 4.315992121815807151158520717919, 5.507281204133779648746489000265, 6.42455242543440210775413199361, 7.34631774815741734154093036173, 7.91788522631816849719678060531, 9.457959391995327752004316590555, 10.00415798875657514998778362966, 11.4619110556134249723576943573, 12.32464665141372619949809899954, 13.367040226824927157442747356515, 14.26762619486831315835534723222, 15.02452735622854038616954613943, 15.4989633473220251060131460656, 16.840655704769162575581039677348, 17.51820527063934659762559987927, 18.23100919293364002094343489075, 19.22797244439228884998893799940, 20.25659467336353323556116321937, 21.50803229074946821149432848111, 22.36538055880878941395995149387, 22.58545159626929412693549666044, 23.794723831742040784001543030649, 24.43149330654233151595052601727