| L(s) = 1 | + (1.83 + 0.509i)2-s + (1.37 + 0.832i)4-s + (−3.83 + 0.148i)5-s + (0.365 + 0.0571i)7-s + (−0.508 − 0.538i)8-s + (−7.09 − 1.68i)10-s + (0.814 + 5.96i)11-s + (−3.43 + 5.00i)13-s + (0.639 + 0.290i)14-s + (−2.15 − 4.09i)16-s + (−3.18 + 1.60i)17-s + (0.159 + 2.73i)19-s + (−5.40 − 2.98i)20-s + (−1.54 + 11.3i)22-s + (0.230 − 0.597i)23-s + ⋯ |
| L(s) = 1 | + (1.29 + 0.360i)2-s + (0.689 + 0.416i)4-s + (−1.71 + 0.0665i)5-s + (0.138 + 0.0215i)7-s + (−0.179 − 0.190i)8-s + (−2.24 − 0.531i)10-s + (0.245 + 1.79i)11-s + (−0.951 + 1.38i)13-s + (0.170 + 0.0776i)14-s + (−0.539 − 1.02i)16-s + (−0.772 + 0.388i)17-s + (0.0365 + 0.627i)19-s + (−1.20 − 0.667i)20-s + (−0.330 + 2.41i)22-s + (0.0480 − 0.124i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338262 + 1.16183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338262 + 1.16183i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.83 - 0.509i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (3.83 - 0.148i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-0.365 - 0.0571i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.814 - 5.96i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (3.43 - 5.00i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.18 - 1.60i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.159 - 2.73i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-0.230 + 0.597i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (0.792 + 0.566i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-3.51 + 4.03i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (3.14 + 4.22i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.857 + 3.32i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.41 - 3.09i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.84 - 3.25i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.566 + 3.21i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.88 + 0.768i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (12.5 - 7.55i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-6.29 + 4.49i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-2.46 - 8.24i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (6.59 - 1.56i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-2.56 + 2.51i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.79 - 14.7i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (1.54 - 5.15i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-12.6 - 0.491i)T + (96.7 + 7.51i)T^{2} \) |
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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
−11.15238546921944590631228835925, −9.861302929983900855171801270758, −9.000147534171560731790900122898, −7.70418259370945199966910967311, −7.15632883309707980454685661260, −6.44558963445218499744528674580, −4.90007399777503723695531802634, −4.33904865410943187187733189934, −3.85005631448763961567656039731, −2.28754762884153502784458652483,
0.39622055406606530537209505881, 2.96393967374781155001160036200, 3.35725616403862270483301814402, 4.50159615246806392690071733715, 5.15317530506279099836904629302, 6.27842873351775634749976381141, 7.43251014144656741272926739086, 8.288703216892492350884736801158, 8.906991975332164531893033290720, 10.58998022237976842970574575634
