| L(s) = 1 | + (1.03 − 0.866i)2-s + (−0.0320 + 0.181i)4-s + (−1.55 + 0.565i)5-s + (0.418 + 2.37i)7-s + (1.47 + 2.54i)8-s + (−1.11 + 1.92i)10-s + (−5.58 − 2.03i)11-s + (−2.47 − 2.07i)13-s + (2.48 + 2.08i)14-s + (3.37 + 1.22i)16-s + (−1.5 + 2.59i)17-s + (3.31 + 5.74i)19-s + (−0.0530 − 0.300i)20-s + (−7.52 + 2.73i)22-s + (−0.511 + 2.89i)23-s + ⋯ |
| L(s) = 1 | + (0.729 − 0.612i)2-s + (−0.0160 + 0.0909i)4-s + (−0.694 + 0.252i)5-s + (0.158 + 0.897i)7-s + (0.520 + 0.901i)8-s + (−0.352 + 0.609i)10-s + (−1.68 − 0.612i)11-s + (−0.685 − 0.575i)13-s + (0.665 + 0.558i)14-s + (0.844 + 0.307i)16-s + (−0.363 + 0.630i)17-s + (0.761 + 1.31i)19-s + (−0.0118 − 0.0672i)20-s + (−1.60 + 0.583i)22-s + (−0.106 + 0.604i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.858115 + 0.909549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858115 + 0.909549i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.03 + 0.866i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (1.55 - 0.565i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.418 - 2.37i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (5.58 + 2.03i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.47 + 2.07i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 - 5.74i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.511 - 2.89i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.988 - 0.829i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.102 - 0.579i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.0209 - 0.0362i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.75 + 3.15i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.87 - 1.77i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.648 + 3.67i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-6.90 + 2.51i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.91 - 10.8i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.42 - 1.19i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 4.77i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 + 4.81i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.89 - 2.43i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.05 + 2.56i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (4.07 + 7.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.245 + 0.0892i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.75620144633903073188385055973, −10.12689255193903811622841055263, −8.690954766343476877610700404053, −7.961857984004208874439139464405, −7.45350004983528413848620140932, −5.58875240731818847720817120411, −5.34086009314810063884943542500, −3.93002952876166923136851428472, −3.09474078405467897628033262595, −2.19097937166501610708544338159,
0.48592131081237900324447648139, 2.54151452276696499929114057980, 4.11261971614694264988432304094, 4.73687063464548157442037133760, 5.39815783639535925963525539395, 6.89440199900665794877349743581, 7.28921071161014778979025542898, 8.085998522264783720415728230061, 9.446056557012390321867820671884, 10.18779728569110391023862365250
