| L(s) = 1 | + (0.162 − 0.0591i)2-s + (−1.50 + 1.26i)4-s + (0.648 + 3.67i)5-s + (2.32 + 1.94i)7-s + (−0.343 + 0.594i)8-s + (0.323 + 0.559i)10-s + (0.432 − 2.45i)11-s + (0.718 + 0.261i)13-s + (0.492 + 0.179i)14-s + (0.663 − 3.76i)16-s + (2.31 + 4.00i)17-s + (0.305 − 0.529i)19-s + (−5.63 − 4.73i)20-s + (−0.0748 − 0.424i)22-s + (−4.99 + 4.19i)23-s + ⋯ |
| L(s) = 1 | + (0.114 − 0.0418i)2-s + (−0.754 + 0.633i)4-s + (0.290 + 1.64i)5-s + (0.877 + 0.736i)7-s + (−0.121 + 0.210i)8-s + (0.102 + 0.177i)10-s + (0.130 − 0.739i)11-s + (0.199 + 0.0725i)13-s + (0.131 + 0.0479i)14-s + (0.165 − 0.940i)16-s + (0.560 + 0.970i)17-s + (0.0701 − 0.121i)19-s + (−1.26 − 1.05i)20-s + (−0.0159 − 0.0904i)22-s + (−1.04 + 0.874i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.635538 + 1.26546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635538 + 1.26546i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.162 + 0.0591i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.648 - 3.67i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 1.94i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.432 + 2.45i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.718 - 0.261i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.31 - 4.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.305 + 0.529i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.99 - 4.19i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.15 - 2.24i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.01 + 4.21i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.47 + 4.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.94 - 1.79i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.967 + 5.48i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.848 + 0.711i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 + (-2.05 - 11.6i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.27 - 5.26i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.13 - 0.414i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.45 + 4.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.14 - 3.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.0 + 4.03i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.47 + 3.08i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.76 + 6.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.164 - 0.933i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.78715736337647021221331748806, −9.866573880994295126513521044598, −8.926228885465578911052453951140, −8.046541089256525726709956234217, −7.41310641337891193073492370669, −6.09155102361642059268719458804, −5.51047213106314644449474847413, −4.00665366933738201892528074819, −3.23880577498883384389646832375, −2.08242087981993995256167566743,
0.76344460732520035621483390127, 1.73704447406699338304554622183, 4.00993054631011286708636346932, 4.77865720077543700945367009190, 5.19831446651974815691015006571, 6.35791255540972834967176495203, 7.80901383605697753008064005977, 8.357595972183247266069964997951, 9.422908562415906250739853224006, 9.786989932941252880062367250165
