| L(s) = 1 | + (1.11 − 0.309i)2-s + (−0.574 + 0.346i)4-s + (−2.97 − 0.115i)5-s + (4.35 − 0.681i)7-s + (−2.11 + 2.23i)8-s + (−3.33 + 0.790i)10-s + (−0.228 + 1.67i)11-s + (0.555 + 0.809i)13-s + (4.62 − 2.10i)14-s + (−1.02 + 1.95i)16-s + (0.733 + 0.368i)17-s + (−0.468 + 8.04i)19-s + (1.74 − 0.963i)20-s + (0.263 + 1.92i)22-s + (2.88 + 7.48i)23-s + ⋯ |
| L(s) = 1 | + (0.785 − 0.218i)2-s + (−0.287 + 0.173i)4-s + (−1.32 − 0.0515i)5-s + (1.64 − 0.257i)7-s + (−0.747 + 0.791i)8-s + (−1.05 + 0.249i)10-s + (−0.0688 + 0.504i)11-s + (0.153 + 0.224i)13-s + (1.23 − 0.562i)14-s + (−0.257 + 0.488i)16-s + (0.177 + 0.0893i)17-s + (−0.107 + 1.84i)19-s + (0.390 − 0.215i)20-s + (0.0561 + 0.410i)22-s + (0.602 + 1.56i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35049 + 0.860450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35049 + 0.860450i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.11 + 0.309i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (2.97 + 0.115i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (-4.35 + 0.681i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (0.228 - 1.67i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (-0.555 - 0.809i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.733 - 0.368i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (0.468 - 8.04i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (-2.88 - 7.48i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (-2.38 + 1.70i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (1.81 + 2.08i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (1.78 - 2.39i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (-2.19 - 8.50i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (1.13 - 1.02i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.86 + 2.13i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (1.22 + 6.95i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.23 - 0.913i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (1.90 + 1.14i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (7.25 + 5.18i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (2.04 - 6.83i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (-14.6 - 3.47i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (10.8 + 10.6i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (1.54 - 6.00i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (5.19 + 17.3i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 0.522i)T + (96.7 - 7.51i)T^{2} \) |
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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
−10.95446102836477320280404918705, −9.740754130797531053356741684235, −8.448592204933596627779829310906, −8.012970875632591890884548603101, −7.36554932786645450144687304933, −5.73953633878273622231144072573, −4.80429602769727810877586250548, −4.16032889863781852291340132422, −3.39258958823643269161813833602, −1.64855448895931878419748270619,
0.70551847543815980067025588361, 2.79801724103107405701565754992, 4.05655378208675481332578367593, 4.74057305784342526530250512611, 5.38931810581654658522080635992, 6.72268813347388002859410220487, 7.60448696983116287694939132371, 8.590602296841836995237789012840, 8.959127549274635581023788409763, 10.68953112616592275826788713469
