| 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 | \( 1 + (-3.67 + 1.33i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.37149984674044292491271602621, −9.659241432212485288371583358629, −9.008886339504521614017021434275, −7.72300881661240446542896653630, −6.39741585564678329963751217049, −5.95332593013596036257693295062, −5.54030650392550113103358919537, −4.14684791338315242107953857003, −2.65832921717516611532379786473, −1.59003019231981783883332441344,
1.43703667937789102383478970181, 2.60277388037505361012481497282, 3.82905360015540609018045200664, 4.83795533021789518208528383653, 5.65109881310182232145496816693, 6.76616400847717730892397778451, 7.68271369880411180354147403106, 8.968303506005734082332832279343, 9.314167855791265639250028357834, 10.45049552979046521493965176552
