| L(s) = 1 | + (−1.57 + 0.574i)2-s + (1.45 + 0.940i)3-s + (0.632 − 0.530i)4-s + (−0.196 − 1.11i)5-s + (−2.83 − 0.649i)6-s + (−2.99 − 2.51i)7-s + (0.987 − 1.70i)8-s + (1.23 + 2.73i)9-s + (0.949 + 1.64i)10-s + (−0.324 + 1.84i)11-s + (1.41 − 0.177i)12-s + (0.688 + 0.250i)13-s + (6.17 + 2.24i)14-s + (0.760 − 1.80i)15-s + (−0.862 + 4.89i)16-s + (−0.944 − 1.63i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.406i)2-s + (0.839 + 0.542i)3-s + (0.316 − 0.265i)4-s + (−0.0877 − 0.497i)5-s + (−1.15 − 0.265i)6-s + (−1.13 − 0.949i)7-s + (0.349 − 0.604i)8-s + (0.410 + 0.911i)9-s + (0.300 + 0.519i)10-s + (−0.0979 + 0.555i)11-s + (0.409 − 0.0511i)12-s + (0.190 + 0.0694i)13-s + (1.65 + 0.600i)14-s + (0.196 − 0.465i)15-s + (−0.215 + 1.22i)16-s + (−0.229 − 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.452032 + 0.155659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452032 + 0.155659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.45 - 0.940i)T \) |
| good | 2 | \( 1 + (1.57 - 0.574i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.196 + 1.11i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.99 + 2.51i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.324 - 1.84i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.688 - 0.250i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.944 + 1.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.46 - 3.74i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.99 + 1.81i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 0.861i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.69 + 2.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.68 + 0.614i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.873 + 4.95i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.30 - 1.09i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + (1.95 + 11.0i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.00 - 3.36i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.77 - 0.646i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.09 - 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.6 - 4.22i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 3.99i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0596 - 0.338i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−17.35293925084269077196099575446, −16.30520389239434089377580965790, −15.70409327413887741392575552744, −13.89445308673065878690489974991, −12.81260397149346580597086378930, −10.31703685939222878699905660678, −9.592026069491252578168680347090, −8.352501640956398843040638514632, −7.06025365009650705601697038266, −4.03413317018922101146181010889,
2.77114971237151580849842537785, 6.51266386921068694724170514511, 8.311651096393578910188396666801, 9.199434506444025268088399254937, 10.49105977442757598117568611797, 12.17712864718856500124134216460, 13.53054307025110635066800909428, 14.88601694797514209754264147158, 16.19054223836369165575579832890, 17.87227482064302236619493481864
