| L(s) = 1 | + (−0.614 + 1.42i)2-s + (−0.277 − 0.294i)4-s + (0.184 + 3.17i)5-s + (0.284 + 0.0675i)7-s + (−2.32 + 0.846i)8-s + (−4.63 − 1.68i)10-s + (−4.85 − 3.19i)11-s + (−0.249 − 0.0291i)13-s + (−0.271 + 0.364i)14-s + (0.270 − 4.63i)16-s + (−3.14 + 2.63i)17-s + (−3.55 − 2.98i)19-s + (0.881 − 0.934i)20-s + (7.53 − 4.95i)22-s + (4.46 − 1.05i)23-s + ⋯ |
| L(s) = 1 | + (−0.434 + 1.00i)2-s + (−0.138 − 0.147i)4-s + (0.0826 + 1.41i)5-s + (0.107 + 0.0255i)7-s + (−0.822 + 0.299i)8-s + (−1.46 − 0.532i)10-s + (−1.46 − 0.963i)11-s + (−0.0692 − 0.00809i)13-s + (−0.0724 + 0.0973i)14-s + (0.0675 − 1.15i)16-s + (−0.762 + 0.639i)17-s + (−0.816 − 0.685i)19-s + (0.197 − 0.208i)20-s + (1.60 − 1.05i)22-s + (0.930 − 0.220i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.273831 - 0.454206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.273831 - 0.454206i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.614 - 1.42i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.184 - 3.17i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-0.284 - 0.0675i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (4.85 + 3.19i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.249 + 0.0291i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (3.14 - 2.63i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.55 + 2.98i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-4.46 + 1.05i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-3.05 - 4.10i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (1.69 - 5.67i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.850 - 4.82i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-0.721 - 1.67i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-1.95 + 0.980i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (2.19 + 7.33i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.52 - 6.10i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.68 - 5.05i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-2.13 + 2.26i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (4.08 - 5.48i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (4.71 + 1.71i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.71 - 0.624i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (4.28 - 9.92i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-3.99 + 9.27i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (9.79 - 3.56i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.379 - 6.51i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.79401650898488298571075375974, −10.29188406507843751162752551509, −8.839382822421755735245207400520, −8.359031546555507703455516590828, −7.33830447287950931618951177904, −6.73185112667594854260261683929, −6.02249018576737548761807665265, −4.96275842950097723801568020184, −3.19631139789104590579246466211, −2.57330976286523857285603295343,
0.29114222247993840085395185284, 1.75141170109731583074583278246, 2.67030142742912474995534052430, 4.30027305278469060169953453651, 5.06053120070828127486365018105, 6.10239348818040969834613464248, 7.47755922543310945449371664132, 8.357349852249861939383437772907, 9.225603447029378571376817645198, 9.784680957866208854179113160537
