| 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 | \( 1 + (-2.55 + 2.71i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.70970879469761541678953066528, −9.633081462020484038497584397541, −8.854271578990819312250320835918, −8.483855259029942122506159443019, −7.33791415315280151611334753063, −6.53595549025861208299660909380, −4.89145741287849404878441084131, −4.04202759163170981427389325392, −2.37210293768974581516236167295, −1.57978425520067746160046752005,
0.25146857408648893231015255635, 1.96402294983123802078533699062, 3.63389046251110589960496583981, 5.02838816417517235651832501096, 6.09902248188461463317376075690, 6.89573001253631139947502671905, 7.78548404370275418124745575165, 8.495984639013962504183846014431, 8.977940071943652680624463537990, 10.11400495554785037978405867616
