| L(s) = 1 | + (−0.463 + 0.304i)2-s + (−0.670 + 1.55i)4-s + (0.355 − 0.377i)5-s + (−2.24 − 0.262i)7-s + (−0.355 − 2.01i)8-s + (−0.0499 + 0.283i)10-s + (0.352 − 1.17i)11-s + (0.226 − 3.88i)13-s + (1.12 − 0.562i)14-s + (−1.54 − 1.63i)16-s + (6.74 − 2.45i)17-s + (1.31 + 0.478i)19-s + (0.347 + 0.806i)20-s + (0.195 + 0.652i)22-s + (1.91 − 0.223i)23-s + ⋯ |
| L(s) = 1 | + (−0.327 + 0.215i)2-s + (−0.335 + 0.777i)4-s + (0.159 − 0.168i)5-s + (−0.848 − 0.0992i)7-s + (−0.125 − 0.712i)8-s + (−0.0157 + 0.0895i)10-s + (0.106 − 0.354i)11-s + (0.0628 − 1.07i)13-s + (0.299 − 0.150i)14-s + (−0.386 − 0.409i)16-s + (1.63 − 0.595i)17-s + (0.301 + 0.109i)19-s + (0.0777 + 0.180i)20-s + (0.0416 + 0.139i)22-s + (0.398 − 0.0466i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04614 - 0.115490i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04614 - 0.115490i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.463 - 0.304i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (-0.355 + 0.377i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (2.24 + 0.262i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.352 + 1.17i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.226 + 3.88i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-6.74 + 2.45i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 0.478i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.91 + 0.223i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (-6.18 - 3.10i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (4.11 + 5.53i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (-3.62 - 3.04i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-1.26 - 0.829i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (2.33 - 0.552i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-6.75 + 9.06i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (3.04 - 5.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.69 + 12.3i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-2.26 - 5.24i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-12.0 + 6.03i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (0.376 - 2.13i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.161 - 0.914i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-6.48 + 4.26i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (-3.50 + 2.30i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (2.46 + 14.0i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (10.6 + 11.3i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−9.994709703049380239042415258618, −9.576352880711897154181036440881, −8.605475927849461609131650831953, −7.78923007316316990815373330070, −7.08992910647328451123532647513, −5.95272313476475038000766296646, −5.01640156209347876786797133858, −3.52545254891426609122243111821, −3.04701131875444330270800020829, −0.74817144662652373475004480254,
1.18431187051565008093219200183, 2.57109070530745697249309917513, 3.91790530989494594715763890903, 5.07326635278254888025638103957, 6.06660351401696871008215639598, 6.73998569916190298951726870380, 7.965649770834617460414754309466, 9.008564138244229821018443992432, 9.657522162381701245241756815614, 10.21814197418247156691891344315
