| L(s) = 1 | + (−1.81 + 1.19i)2-s + (1.08 − 2.51i)4-s + (−0.443 + 0.470i)5-s + (1.81 + 0.212i)7-s + (0.277 + 1.57i)8-s + (0.244 − 1.38i)10-s + (0.346 − 1.15i)11-s + (0.310 − 5.32i)13-s + (−3.55 + 1.78i)14-s + (1.36 + 1.44i)16-s + (−6.43 + 2.34i)17-s + (−5.97 − 2.17i)19-s + (0.700 + 1.62i)20-s + (0.754 + 2.52i)22-s + (3.09 − 0.361i)23-s + ⋯ |
| L(s) = 1 | + (−1.28 + 0.845i)2-s + (0.541 − 1.25i)4-s + (−0.198 + 0.210i)5-s + (0.686 + 0.0802i)7-s + (0.0980 + 0.556i)8-s + (0.0772 − 0.438i)10-s + (0.104 − 0.349i)11-s + (0.0860 − 1.47i)13-s + (−0.950 + 0.477i)14-s + (0.341 + 0.362i)16-s + (−1.56 + 0.567i)17-s + (−1.37 − 0.499i)19-s + (0.156 + 0.362i)20-s + (0.160 + 0.537i)22-s + (0.645 − 0.0754i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.249339 - 0.207010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.249339 - 0.207010i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.81 - 1.19i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (0.443 - 0.470i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-1.81 - 0.212i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.346 + 1.15i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.310 + 5.32i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (6.43 - 2.34i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (5.97 + 2.17i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 0.361i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (5.26 + 2.64i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (1.65 + 2.22i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (1.09 + 0.918i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (0.931 + 0.612i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (9.37 - 2.22i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-3.64 + 4.89i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (-4.26 + 7.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.598 + 2.00i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (1.42 + 3.29i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-1.09 + 0.547i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.41 + 8.02i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 6.32i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-11.9 + 7.83i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (5.61 - 3.69i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (2.70 + 15.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.55 - 2.71i)T + (-5.64 + 96.8i)T^{2} \) |
| 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
−10.11400495554785037978405867616, −8.977940071943652680624463537990, −8.495984639013962504183846014431, −7.78548404370275418124745575165, −6.89573001253631139947502671905, −6.09902248188461463317376075690, −5.02838816417517235651832501096, −3.63389046251110589960496583981, −1.96402294983123802078533699062, −0.25146857408648893231015255635,
1.57978425520067746160046752005, 2.37210293768974581516236167295, 4.04202759163170981427389325392, 4.89145741287849404878441084131, 6.53595549025861208299660909380, 7.33791415315280151611334753063, 8.483855259029942122506159443019, 8.854271578990819312250320835918, 9.633081462020484038497584397541, 10.70970879469761541678953066528
