| L(s) = 1 | + (−1.29 − 0.848i)2-s + (0.152 + 0.354i)4-s + (0.349 + 0.370i)5-s + (−3.96 + 0.463i)7-s + (−0.432 + 2.45i)8-s + (−0.136 − 0.774i)10-s + (1.09 + 3.66i)11-s + (−0.181 − 3.10i)13-s + (5.50 + 2.76i)14-s + (3.17 − 3.36i)16-s + (−1.41 − 0.513i)17-s + (6.30 − 2.29i)19-s + (−0.0778 + 0.180i)20-s + (1.69 − 5.66i)22-s + (1.17 + 0.137i)23-s + ⋯ |
| L(s) = 1 | + (−0.912 − 0.600i)2-s + (0.0764 + 0.177i)4-s + (0.156 + 0.165i)5-s + (−1.49 + 0.175i)7-s + (−0.153 + 0.868i)8-s + (−0.0431 − 0.244i)10-s + (0.331 + 1.10i)11-s + (−0.0502 − 0.862i)13-s + (1.47 + 0.738i)14-s + (0.793 − 0.840i)16-s + (−0.342 − 0.124i)17-s + (1.44 − 0.526i)19-s + (−0.0173 + 0.0403i)20-s + (0.361 − 1.20i)22-s + (0.245 + 0.0287i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.577240 - 0.410600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577240 - 0.410600i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.29 + 0.848i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-0.349 - 0.370i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (3.96 - 0.463i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 3.66i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (0.181 + 3.10i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (1.41 + 0.513i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-6.30 + 2.29i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 0.137i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-6.17 + 3.10i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (0.0276 - 0.0371i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (2.33 - 1.96i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-4.96 + 3.26i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-0.231 - 0.0549i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (2.82 + 3.80i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-6.81 - 11.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.400 + 1.33i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (0.124 - 0.288i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (-4.90 - 2.46i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (2.03 + 11.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.70 + 15.3i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-11.6 - 7.69i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.386i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-2.00 + 11.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.92 + 9.46i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.06986251273376368386458153063, −9.545160820612522117273584857005, −8.906394101880943073354264445331, −7.74592950064278293478759835626, −6.81365913345522778426581969265, −5.87839612036789466636208568986, −4.76566705437931413533895344189, −3.20284421553495849932047672267, −2.37436433902276399903524737471, −0.68034864862434686182500903644,
0.957978362628039778226041428453, 3.12117737889392918575579307217, 3.86445366444088374385612445330, 5.51442876694796083393115674679, 6.53766883216132873629842728288, 6.96247975849617207035647453236, 8.077433547483631648061023479813, 8.991226617557949414717542001583, 9.457562219372637542271092173363, 10.14878939121849073566447789385
