| L(s) = 1 | + (−1.00 + 0.662i)2-s + (−0.215 + 0.500i)4-s + (−2.69 + 2.85i)5-s + (−1.84 − 0.215i)7-s + (−0.533 − 3.02i)8-s + (0.823 − 4.66i)10-s + (−1.39 + 4.64i)11-s + (0.0180 − 0.310i)13-s + (2.00 − 1.00i)14-s + (1.79 + 1.89i)16-s + (−3.17 + 1.15i)17-s + (1.05 + 0.385i)19-s + (−0.848 − 1.96i)20-s + (−1.67 − 5.60i)22-s + (2.26 − 0.264i)23-s + ⋯ |
| L(s) = 1 | + (−0.712 + 0.468i)2-s + (−0.107 + 0.250i)4-s + (−1.20 + 1.27i)5-s + (−0.698 − 0.0815i)7-s + (−0.188 − 1.06i)8-s + (0.260 − 1.47i)10-s + (−0.419 + 1.40i)11-s + (0.00501 − 0.0861i)13-s + (0.535 − 0.268i)14-s + (0.448 + 0.474i)16-s + (−0.770 + 0.280i)17-s + (0.243 + 0.0885i)19-s + (−0.189 − 0.440i)20-s + (−0.357 − 1.19i)22-s + (0.471 − 0.0551i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0305052 - 0.0215044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0305052 - 0.0215044i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.00 - 0.662i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (2.69 - 2.85i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (1.84 + 0.215i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (1.39 - 4.64i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.0180 + 0.310i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (3.17 - 1.15i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 0.385i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.26 + 0.264i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (2.56 + 1.28i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (2.90 + 3.89i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (-8.69 - 7.29i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 0.793i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (-4.35 + 1.03i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-2.34 + 3.15i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (-0.812 + 1.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.86 + 9.57i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (0.302 + 0.702i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-0.658 + 0.330i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (1.18 - 6.69i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (0.692 + 3.92i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (10.3 - 6.82i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (-6.18 + 4.06i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (0.943 + 5.35i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (3.16 + 3.34i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.95366068302299709988020822629, −10.03676940267416283097118064358, −9.447965862247347624320311476772, −8.250289924929009027280455330406, −7.53606564932681376265388800985, −7.03319245179179498782553883793, −6.30453466883956013602520966816, −4.47325842250590405081849878995, −3.65892644859644965223040446602, −2.63548567688260698765166120714,
0.03051996452757256338799353936, 1.02525864190456857669612893327, 2.85299153480519459567942622913, 4.07153460972687131484565890978, 5.13489556325279657690144164705, 5.96410820714409958591832011921, 7.44825870559803892682463338880, 8.271499128569847997864565987860, 9.059310831563779309505392127217, 9.310496912517493369642377997527
