| L(s) = 1 | + (−0.588 − 1.36i)2-s + (−0.141 + 0.149i)4-s + (−0.0932 + 1.60i)5-s + (−1.85 + 0.438i)7-s + (−2.50 − 0.911i)8-s + (2.23 − 0.814i)10-s + (−3.38 + 2.22i)11-s + (1.67 − 0.196i)13-s + (1.68 + 2.26i)14-s + (0.254 + 4.36i)16-s + (4.14 + 3.47i)17-s + (3.70 − 3.10i)19-s + (−0.226 − 0.240i)20-s + (5.03 + 3.30i)22-s + (0.651 + 0.154i)23-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.964i)2-s + (−0.0706 + 0.0749i)4-s + (−0.0416 + 0.715i)5-s + (−0.699 + 0.165i)7-s + (−0.885 − 0.322i)8-s + (0.707 − 0.257i)10-s + (−1.02 + 0.671i)11-s + (0.465 − 0.0544i)13-s + (0.450 + 0.605i)14-s + (0.0635 + 1.09i)16-s + (1.00 + 0.844i)17-s + (0.849 − 0.712i)19-s + (−0.0506 − 0.0537i)20-s + (1.07 + 0.705i)22-s + (0.135 + 0.0322i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864663 + 0.166255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864663 + 0.166255i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.588 + 1.36i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (0.0932 - 1.60i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (1.85 - 0.438i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (3.38 - 2.22i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 0.196i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-4.14 - 3.47i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.70 + 3.10i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.651 - 0.154i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (2.85 - 3.83i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (-1.39 - 4.66i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.07 - 11.7i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (4.00 - 9.28i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-7.64 - 3.83i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.0578 + 0.193i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-2.48 + 4.29i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.58 - 1.04i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-0.124 - 0.131i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (7.95 + 10.6i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-9.41 + 3.42i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (10.9 + 3.97i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.45 + 3.37i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (2.17 + 5.03i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (6.65 + 2.42i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.0778 - 1.33i)T + (-96.3 + 11.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36378982262180167537121534760, −9.935282999038440078815763604937, −9.042292409319618452748003112548, −7.954759934386691032698618632273, −6.90799151240319035122018373559, −6.13354548427263973850228833685, −4.99519101784562465543139386773, −3.25047554493592004060208978339, −2.93303807384907315832721379395, −1.42285862591785635012445385268,
0.54039497531238048568727637539, 2.73184848105866042909409875899, 3.80319990910919190793901709146, 5.55467083912137662410966473007, 5.67019045244194233922340216631, 7.09107493896957383112846086466, 7.67762094949207735417197374541, 8.515393224019527114505136673419, 9.249963019731520542684531132052, 10.07078129070551007336399055963
