| L(s) = 1 | + (−1.17 − 0.327i)2-s + (−0.436 − 0.263i)4-s + (−3.30 + 0.128i)5-s + (3.02 + 0.473i)7-s + (2.10 + 2.22i)8-s + (3.93 + 0.931i)10-s + (−0.345 − 2.53i)11-s + (−2.19 + 3.20i)13-s + (−3.40 − 1.54i)14-s + (−1.26 − 2.40i)16-s + (2.28 − 1.14i)17-s + (−0.0846 − 1.45i)19-s + (1.47 + 0.815i)20-s + (−0.421 + 3.08i)22-s + (−1.73 + 4.50i)23-s + ⋯ |
| L(s) = 1 | + (−0.831 − 0.231i)2-s + (−0.218 − 0.131i)4-s + (−1.47 + 0.0574i)5-s + (1.14 + 0.179i)7-s + (0.743 + 0.787i)8-s + (1.24 + 0.294i)10-s + (−0.104 − 0.762i)11-s + (−0.609 + 0.889i)13-s + (−0.910 − 0.413i)14-s + (−0.316 − 0.601i)16-s + (0.554 − 0.278i)17-s + (−0.0194 − 0.333i)19-s + (0.330 + 0.182i)20-s + (−0.0899 + 0.658i)22-s + (−0.362 + 0.938i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.286204 - 0.394580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.286204 - 0.394580i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.17 + 0.327i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (3.30 - 0.128i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-3.02 - 0.473i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.345 + 2.53i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (2.19 - 3.20i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.28 + 1.14i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.0846 + 1.45i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (1.73 - 4.50i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-4.11 - 2.93i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-5.18 + 5.94i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (5.96 + 8.00i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-1.91 + 7.44i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (8.37 + 7.59i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (4.91 + 5.63i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.683 + 3.87i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (1.90 - 0.779i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-7.42 + 4.48i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (6.69 - 4.78i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (2.48 + 8.30i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (2.45 - 0.580i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-7.41 + 7.27i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.15 - 8.37i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-4.79 + 16.0i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (6.93 + 0.269i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.18521316338281738104707735858, −9.086281280574294614412825828366, −8.420873158718039609813481954378, −7.81796582389890312852265453994, −7.08627896875105066944848056429, −5.39502625817300408044635991795, −4.64158021313711344656353644235, −3.61930044749731403332477605842, −1.94977569553314281478881398343, −0.40609026652720932437522199889,
1.18420142394264700803961027818, 3.17418248960914301456749072212, 4.50326707855559487089710024522, 4.78965957900881878445286014652, 6.66791889759925137391174803357, 7.66438099755273665041415872609, 8.099773558144518434306771877352, 8.459807329517686250543060934862, 9.936114725804616092652530594512, 10.36946986248341392378204183661
