| L(s) = 1 | + (−1.74 − 1.84i)2-s + (−0.258 + 4.44i)4-s + (3.16 − 0.369i)5-s + (−1.48 − 0.745i)7-s + (4.77 − 4.00i)8-s + (−6.20 − 5.20i)10-s + (0.777 + 1.80i)11-s + (−3.77 − 0.894i)13-s + (1.20 + 4.04i)14-s + (−6.88 − 0.804i)16-s + (0.757 − 4.29i)17-s + (0.245 + 1.39i)19-s + (0.825 + 14.1i)20-s + (1.97 − 4.57i)22-s + (2.02 − 1.01i)23-s + ⋯ |
| L(s) = 1 | + (−1.23 − 1.30i)2-s + (−0.129 + 2.22i)4-s + (1.41 − 0.165i)5-s + (−0.560 − 0.281i)7-s + (1.68 − 1.41i)8-s + (−1.96 − 1.64i)10-s + (0.234 + 0.543i)11-s + (−1.04 − 0.248i)13-s + (0.323 + 1.08i)14-s + (−1.72 − 0.201i)16-s + (0.183 − 1.04i)17-s + (0.0564 + 0.320i)19-s + (0.184 + 3.16i)20-s + (0.420 − 0.975i)22-s + (0.422 − 0.212i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.402541 - 0.787070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402541 - 0.787070i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.74 + 1.84i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-3.16 + 0.369i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (1.48 + 0.745i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.777 - 1.80i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (3.77 + 0.894i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.757 + 4.29i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.245 - 1.39i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.02 + 1.01i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-2.70 + 9.04i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-3.49 - 2.29i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-6.43 + 2.34i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.44 + 2.59i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-4.45 + 5.98i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-5.79 + 3.81i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 2.34i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.40 - 3.24i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.162 - 2.78i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-0.528 - 1.76i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-1.75 - 1.47i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (8.76 - 7.35i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (4.78 + 5.06i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-0.0736 - 0.0780i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-4.04 + 3.39i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.32 - 0.505i)T + (94.3 + 22.3i)T^{2} \) |
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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
−9.937665966152088605006279343361, −9.603427326007573259330501325892, −8.879247228992268978325272674297, −7.70705689072121852126418333019, −6.87195171135077917406082516467, −5.60369675368182859833129858881, −4.29216321110987735799559266708, −2.79777638749303355030258925934, −2.18597531830374055754796699949, −0.76005498702077691718927795025,
1.31450162613456750347925765323, 2.75269749259864680379649211626, 4.86186909219588181258812932471, 5.92426881842990380824243078455, 6.27631655150549547982243221309, 7.15718483573120736300543545595, 8.152729454375310414958979164600, 9.161334147314763080441346855432, 9.485468314911685846194449175915, 10.24684566534546143988649635236
