| L(s) = 1 | + (−0.0800 − 0.185i)2-s + (1.34 − 1.42i)4-s + (−0.0529 + 0.909i)5-s + (0.159 − 0.0378i)7-s + (−0.751 − 0.273i)8-s + (0.173 − 0.0629i)10-s + (0.626 − 0.412i)11-s + (3.50 − 0.410i)13-s + (−0.0197 − 0.0265i)14-s + (−0.218 − 3.75i)16-s + (−3.09 − 2.59i)17-s + (1.63 − 1.36i)19-s + (1.22 + 1.29i)20-s + (−0.126 − 0.0832i)22-s + (7.98 + 1.89i)23-s + ⋯ |
| L(s) = 1 | + (−0.0566 − 0.131i)2-s + (0.672 − 0.712i)4-s + (−0.0236 + 0.406i)5-s + (0.0603 − 0.0142i)7-s + (−0.265 − 0.0967i)8-s + (0.0547 − 0.0199i)10-s + (0.188 − 0.124i)11-s + (0.973 − 0.113i)13-s + (−0.00529 − 0.00710i)14-s + (−0.0546 − 0.937i)16-s + (−0.750 − 0.629i)17-s + (0.374 − 0.314i)19-s + (0.273 + 0.290i)20-s + (−0.0269 − 0.0177i)22-s + (1.66 + 0.394i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62537 - 0.733256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62537 - 0.733256i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0800 + 0.185i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (0.0529 - 0.909i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-0.159 + 0.0378i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-0.626 + 0.412i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-3.50 + 0.410i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (3.09 + 2.59i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 1.36i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-7.98 - 1.89i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (3.92 - 5.27i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (1.60 + 5.36i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.783 + 4.44i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-4.33 + 10.0i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (3.09 + 1.55i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (2.56 - 8.55i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.06 + 5.31i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.43 - 1.60i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-4.84 - 5.13i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-5.30 - 7.12i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (8.94 - 3.25i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.28 - 3.01i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.27 + 2.96i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (6.66 + 15.4i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 3.90i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.263 - 4.52i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−10.52449607040115856755647334551, −9.361294623663389926647094750499, −8.845319716948007428507530829811, −7.35158807535160377978013482211, −6.86692163195159365562718319882, −5.85918370615409141297531123666, −5.00361829827814504085601238760, −3.52735729377884544749045870543, −2.48560472636656161971123343999, −1.07551817724870831111593116827,
1.50576298816225960380972645332, 2.93148604240181183419753616125, 3.94330544727674584197995420095, 5.07696389175990890053841772847, 6.36598362559362064861608589729, 6.88607660473472970582611781755, 8.092935673006315878601469814351, 8.588345565322517355912380079487, 9.489782799236695485785875336477, 10.79133751418193191834407968751
