| L(s) = 1 | + (−0.837 + 1.94i)2-s + (−1.69 − 1.79i)4-s + (−0.0261 − 0.448i)5-s + (−2.37 − 0.562i)7-s + (0.937 − 0.341i)8-s + (0.892 + 0.324i)10-s + (2.86 + 1.88i)11-s + (−3.02 − 0.353i)13-s + (3.08 − 4.13i)14-s + (0.164 − 2.82i)16-s + (−1.96 + 1.65i)17-s + (−5.90 − 4.95i)19-s + (−0.761 + 0.807i)20-s + (−6.05 + 3.98i)22-s + (5.55 − 1.31i)23-s + ⋯ |
| L(s) = 1 | + (−0.592 + 1.37i)2-s + (−0.848 − 0.898i)4-s + (−0.0116 − 0.200i)5-s + (−0.897 − 0.212i)7-s + (0.331 − 0.120i)8-s + (0.282 + 0.102i)10-s + (0.862 + 0.567i)11-s + (−0.838 − 0.0979i)13-s + (0.823 − 1.10i)14-s + (0.0412 − 0.707i)16-s + (−0.477 + 0.400i)17-s + (−1.35 − 1.13i)19-s + (−0.170 + 0.180i)20-s + (−1.29 + 0.848i)22-s + (1.15 − 0.274i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.474469 - 0.104834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474469 - 0.104834i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.837 - 1.94i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.0261 + 0.448i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (2.37 + 0.562i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 1.88i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (3.02 + 0.353i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (1.96 - 1.65i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (5.90 + 4.95i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-5.55 + 1.31i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (1.87 + 2.51i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 8.01i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.0238 - 0.135i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (4.79 + 11.1i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-4.05 + 2.03i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (0.745 + 2.48i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (0.184 + 0.319i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.82 - 5.80i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.55 + 1.64i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.831 + 1.11i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (3.35 + 1.22i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.75 + 1.73i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (4.06 - 9.43i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-0.387 + 0.898i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-8.14 + 2.96i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.123 + 2.12i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−9.959311009074905413683239169350, −9.135183371321407995571137256419, −8.742475574050561990825392478224, −7.53406541014649211032370602864, −6.77286113704621902035971895936, −6.38638732936476314881670714264, −5.11272212415134944026886671800, −4.15659861253025885312903054289, −2.53606378453374916280480636883, −0.31448663702065229738399394185,
1.39601506024774097457747642364, 2.78375529082772980547278277940, 3.43687724683066646680881325944, 4.68653485284940558405556463483, 6.22752641696589635509802108698, 6.86189139036732215101630755480, 8.320615801960084423752960799385, 9.091579786716014441977362701911, 9.630772917029952922918006248678, 10.54403928816819036083923742257
