| L(s) = 1 | + (1.76 + 1.16i)2-s + (0.978 + 2.26i)4-s + (2.05 + 2.17i)5-s + (3.48 − 0.407i)7-s + (−0.172 + 0.977i)8-s + (1.09 + 6.22i)10-s + (−0.562 − 1.87i)11-s + (−0.0969 − 1.66i)13-s + (6.63 + 3.33i)14-s + (1.94 − 2.06i)16-s + (−3.65 − 1.33i)17-s + (−0.0155 + 0.00566i)19-s + (−2.92 + 6.77i)20-s + (1.18 − 3.97i)22-s + (−7.67 − 0.896i)23-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.821i)2-s + (0.489 + 1.13i)4-s + (0.917 + 0.972i)5-s + (1.31 − 0.154i)7-s + (−0.0609 + 0.345i)8-s + (0.347 + 1.96i)10-s + (−0.169 − 0.566i)11-s + (−0.0268 − 0.461i)13-s + (1.77 + 0.890i)14-s + (0.487 − 0.516i)16-s + (−0.886 − 0.322i)17-s + (−0.00357 + 0.00130i)19-s + (−0.653 + 1.51i)20-s + (0.253 − 0.846i)22-s + (−1.59 − 0.186i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.90903 + 2.18055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.90903 + 2.18055i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.76 - 1.16i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-2.05 - 2.17i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.48 + 0.407i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (0.562 + 1.87i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (0.0969 + 1.66i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (3.65 + 1.33i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.0155 - 0.00566i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (7.67 + 0.896i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (4.12 - 2.07i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (6.12 - 8.23i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-5.48 + 4.60i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (4.72 - 3.10i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (4.44 + 1.05i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (-1.71 - 2.30i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-1.40 - 2.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.42 - 4.74i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 8.61i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (4.00 + 2.01i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.33 - 7.58i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.696 - 3.94i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (6.81 + 4.47i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-4.30 - 2.83i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (0.829 - 4.70i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.98 + 8.45i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.73945137995257580389971547706, −9.874074219477110808821373318758, −8.574605349921073671740548858884, −7.62962939905696625425062390813, −6.85959748768618010337062065247, −5.96140427764854824605575768910, −5.34312489579644315701888724267, −4.37330952093350331472198154828, −3.22220748215648777350462555559, −1.97626963884506027394356732174,
1.83772004359745919892174471039, 2.06158880431061492125213592608, 3.99178397494148144373001826183, 4.65235880502354717347419114724, 5.39969800923300329164676408930, 6.13960359532213676783102937433, 7.72025467555739117890057532673, 8.584086810634648265803862776504, 9.543583527620240516166685997305, 10.41203944564544198606574250213
