| L(s) = 1 | + (−1.63 − 0.0211i)2-s + (0.989 − 1.42i)3-s + (0.666 + 0.0172i)4-s + (−1.81 − 0.557i)5-s + (−1.64 + 2.30i)6-s + (−0.319 + 0.796i)7-s + (2.17 + 0.0844i)8-s + (−1.04 − 2.81i)9-s + (2.95 + 0.948i)10-s + (3.32 + 1.89i)11-s + (0.683 − 0.930i)12-s + (−2.56 − 5.73i)13-s + (0.538 − 1.29i)14-s + (−2.59 + 2.03i)15-s + (−4.88 − 0.252i)16-s + (4.27 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.0149i)2-s + (0.571 − 0.820i)3-s + (0.333 + 0.00861i)4-s + (−0.813 − 0.249i)5-s + (−0.671 + 0.939i)6-s + (−0.120 + 0.301i)7-s + (0.769 + 0.0298i)8-s + (−0.347 − 0.937i)9-s + (0.935 + 0.299i)10-s + (1.00 + 0.570i)11-s + (0.197 − 0.268i)12-s + (−0.710 − 1.59i)13-s + (0.143 − 0.345i)14-s + (−0.668 + 0.525i)15-s + (−1.22 − 0.0631i)16-s + (1.03 + 0.470i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0465492 - 0.462518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0465492 - 0.462518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 + 1.42i)T \) |
| good | 2 | \( 1 + (1.63 + 0.0211i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (1.81 + 0.557i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (0.319 - 0.796i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (-3.32 - 1.89i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (2.56 + 5.73i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (-4.27 - 1.94i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.598 + 0.872i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (-1.80 + 5.49i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (4.75 - 3.03i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.863 + 3.17i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (9.86 + 1.93i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (7.78 - 2.94i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (-3.86 + 1.69i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (0.341 - 1.25i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (2.97 - 3.15i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-7.33 + 1.24i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (2.57 - 4.73i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (1.93 - 1.00i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (11.3 + 8.78i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (7.66 - 2.45i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (0.586 - 3.08i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (0.888 + 5.44i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (11.9 + 4.86i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (-2.66 + 11.5i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−9.813747631885739752907237431004, −8.884899333380532176081286888841, −8.363777643732517125059909514949, −7.55729821514645151327513063264, −7.06665682856597207394250680363, −5.69073453066931984073289732697, −4.29629890316215298287408795351, −3.09659103700823732679741379888, −1.64641513157476747207273009181, −0.34427040859517649593740870391,
1.70338276698036102546921343322, 3.48886277410347820989208803958, 4.08701437908205083186878935425, 5.24796122587184655917386361317, 6.97492863398299247927210825151, 7.48648707009639818197480933608, 8.463700811993833236719305451310, 9.174003034986268033120901294489, 9.691624210037749555358005587014, 10.49127199434299588977776312399
