| L(s) = 1 | + (−1.04 − 0.290i)2-s + (1.59 + 0.680i)3-s + (−0.704 − 0.425i)4-s + (−2.06 + 0.0803i)5-s + (−1.46 − 1.17i)6-s + (−4.16 − 0.651i)7-s + (2.10 + 2.22i)8-s + (2.07 + 2.16i)9-s + (2.18 + 0.518i)10-s + (0.767 + 5.61i)11-s + (−0.832 − 1.15i)12-s + (−3.39 + 4.94i)13-s + (4.16 + 1.89i)14-s + (−3.35 − 1.28i)15-s + (−0.781 − 1.48i)16-s + (1.18 − 0.593i)17-s + ⋯ |
| L(s) = 1 | + (−0.739 − 0.205i)2-s + (0.919 + 0.393i)3-s + (−0.352 − 0.212i)4-s + (−0.925 + 0.0359i)5-s + (−0.598 − 0.479i)6-s + (−1.57 − 0.246i)7-s + (0.743 + 0.787i)8-s + (0.691 + 0.722i)9-s + (0.691 + 0.163i)10-s + (0.231 + 1.69i)11-s + (−0.240 − 0.333i)12-s + (−0.940 + 1.37i)13-s + (1.11 + 0.505i)14-s + (−0.865 − 0.330i)15-s + (−0.195 − 0.371i)16-s + (0.286 − 0.144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.204220 + 0.371768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204220 + 0.371768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.59 - 0.680i)T \) |
| good | 2 | \( 1 + (1.04 + 0.290i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.06 - 0.0803i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (4.16 + 0.651i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.767 - 5.61i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (3.39 - 4.94i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.18 + 0.593i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.383 + 6.58i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.756 - 1.96i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (3.30 + 2.36i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-0.0367 + 0.0420i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (2.13 + 2.87i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (0.587 - 2.28i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-2.40 - 2.18i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.143 - 0.164i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (1.06 - 6.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.683 + 0.279i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 6.83i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (4.99 - 3.56i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-3.11 - 10.4i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (1.60 - 0.380i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (0.413 - 0.405i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.994 + 3.85i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (2.78 - 9.29i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (6.85 + 0.266i)T + (96.7 + 7.51i)T^{2} \) |
| 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
−12.46025532125764623856046636311, −11.30533532543545103582561470530, −9.884242668854651048891741245457, −9.661828939893860977579842919325, −8.930505399813407226714076567129, −7.49069378598291942738394051886, −7.01531788928018497600630207018, −4.71081220790856322987259240498, −3.94567234426126922305690474518, −2.30835225055899918591982724451,
0.38358322038510977996197886414, 3.22814688938447606026853290894, 3.71352479850948983649169385522, 5.91819902300502728724547630543, 7.19809482939450021709592012432, 8.077634787255274811209489957558, 8.613844924427800566289074763152, 9.676355209623997362562784593427, 10.37943944614769401384571494483, 12.13286050642108357980349232571
