| L(s) = 1 | + (−1.90 − 0.0246i)2-s + (1.72 − 0.0902i)3-s + (1.64 + 0.0424i)4-s + (0.325 + 0.0997i)5-s + (−3.30 + 0.129i)6-s + (0.868 − 2.16i)7-s + (0.682 + 0.0264i)8-s + (2.98 − 0.312i)9-s + (−0.618 − 0.198i)10-s + (−5.40 − 3.07i)11-s + (2.84 − 0.0747i)12-s + (−1.97 − 4.42i)13-s + (−1.71 + 4.11i)14-s + (0.571 + 0.143i)15-s + (−4.58 − 0.237i)16-s + (−4.50 − 2.04i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 − 0.0174i)2-s + (0.998 − 0.0521i)3-s + (0.820 + 0.0212i)4-s + (0.145 + 0.0446i)5-s + (−1.34 + 0.0529i)6-s + (0.328 − 0.818i)7-s + (0.241 + 0.00935i)8-s + (0.994 − 0.104i)9-s + (−0.195 − 0.0627i)10-s + (−1.62 − 0.926i)11-s + (0.820 − 0.0215i)12-s + (−0.548 − 1.22i)13-s + (−0.457 + 1.09i)14-s + (0.147 + 0.0369i)15-s + (−1.14 − 0.0592i)16-s + (−1.09 − 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.347752 - 0.621994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.347752 - 0.621994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 + 0.0902i)T \) |
| good | 2 | \( 1 + (1.90 + 0.0246i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (-0.325 - 0.0997i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (-0.868 + 2.16i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (5.40 + 3.07i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (1.97 + 4.42i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (4.50 + 2.04i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.34 - 3.41i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (0.672 - 2.05i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (5.39 - 3.45i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.34 + 4.94i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (-1.42 - 0.279i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (1.31 - 0.499i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (-7.05 + 3.09i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (2.79 - 10.2i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (-7.20 + 7.63i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (6.97 - 1.18i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (1.34 - 2.48i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (0.0407 - 0.0211i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (-1.52 - 1.18i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (-0.476 + 0.152i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (-3.05 + 16.0i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (1.35 + 8.33i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (-4.15 - 1.69i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (-2.50 + 10.8i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.04189505002614440636131703977, −9.230540662161267924919603130243, −8.294320550416323972894565550087, −7.64746136585213677806670397679, −7.41665272638884262267169961707, −5.72658839540830858382656376081, −4.48240465287327577888207126483, −3.12552975375554706109439817359, −2.04645316796996993592104969769, −0.47670191691132106256422219029,
2.01522974404640310686295408060, 2.35847006926760093453418225079, 4.28015189164622663543593039505, 5.17169120854062276427869713181, 6.86600772844509974486289115037, 7.52953288605839127142006522227, 8.257677137365645743559381308695, 9.111979076559572812880341645895, 9.473775809260168952289426346384, 10.34329347532418473346898385244
