| L(s) = 1 | + (0.673 − 0.565i)2-s + (−0.213 + 1.20i)4-s + (3.64 − 1.32i)5-s + (−0.379 − 2.15i)7-s + (1.41 + 2.45i)8-s + (1.70 − 2.95i)10-s + (0.152 + 0.0555i)11-s + (1.84 + 1.55i)13-s + (−1.47 − 1.23i)14-s + (0.0393 + 0.0143i)16-s + (1.5 − 2.59i)17-s + (−1.79 − 3.11i)19-s + (0.826 + 4.68i)20-s + (0.134 − 0.0488i)22-s + (−0.492 + 2.79i)23-s + ⋯ |
| L(s) = 1 | + (0.476 − 0.399i)2-s + (−0.106 + 0.604i)4-s + (1.63 − 0.593i)5-s + (−0.143 − 0.813i)7-s + (0.501 + 0.868i)8-s + (0.539 − 0.934i)10-s + (0.0460 + 0.0167i)11-s + (0.512 + 0.429i)13-s + (−0.393 − 0.330i)14-s + (0.00984 + 0.00358i)16-s + (0.363 − 0.630i)17-s + (−0.412 − 0.714i)19-s + (0.184 + 1.04i)20-s + (0.0286 − 0.0104i)22-s + (−0.102 + 0.582i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.44891 - 0.580403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44891 - 0.580403i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.673 + 0.565i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-3.64 + 1.32i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.379 + 2.15i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.152 - 0.0555i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.84 - 1.55i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.79 + 3.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.492 - 2.79i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.14 - 4.31i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.900 - 5.10i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 5.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.44 - 3.72i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.85 - 2.12i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.28 + 7.28i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + (4.81 - 1.75i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.656 + 3.72i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.49 + 3.76i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.65 - 13.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.34 + 7.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.971 - 0.815i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.49 + 5.44i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.86 - 6.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.67 - 1.33i)T + (74.3 + 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.45049552979046521493965176552, −9.314167855791265639250028357834, −8.968303506005734082332832279343, −7.68271369880411180354147403106, −6.76616400847717730892397778451, −5.65109881310182232145496816693, −4.83795533021789518208528383653, −3.82905360015540609018045200664, −2.60277388037505361012481497282, −1.43703667937789102383478970181,
1.59003019231981783883332441344, 2.65832921717516611532379786473, 4.14684791338315242107953857003, 5.54030650392550113103358919537, 5.95332593013596036257693295062, 6.39741585564678329963751217049, 7.72300881661240446542896653630, 9.008886339504521614017021434275, 9.659241432212485288371583358629, 10.37149984674044292491271602621
