| L(s) = 1 | + (11.7 + 4.27i)2-s + (−44.8 + 13.3i)3-s + (21.3 + 17.9i)4-s + (58.4 − 331. i)5-s + (−582. − 34.4i)6-s + (−23.9 + 20.0i)7-s + (−624. − 1.08e3i)8-s + (1.82e3 − 1.19e3i)9-s + (2.10e3 − 3.63e3i)10-s + (−823. − 4.67e3i)11-s + (−1.19e3 − 517. i)12-s + (−1.31e3 + 479. i)13-s + (−366. + 133. i)14-s + (1.81e3 + 1.56e4i)15-s + (−3.32e3 − 1.88e4i)16-s + (−8.97e3 + 1.55e4i)17-s + ⋯ |
| L(s) = 1 | + (1.03 + 0.377i)2-s + (−0.958 + 0.286i)3-s + (0.166 + 0.140i)4-s + (0.209 − 1.18i)5-s + (−1.10 − 0.0650i)6-s + (−0.0263 + 0.0221i)7-s + (−0.431 − 0.747i)8-s + (0.836 − 0.548i)9-s + (0.664 − 1.15i)10-s + (−0.186 − 1.05i)11-s + (−0.200 − 0.0864i)12-s + (−0.166 + 0.0605i)13-s + (−0.0356 + 0.0129i)14-s + (0.138 + 1.19i)15-s + (−0.203 − 1.15i)16-s + (−0.443 + 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0556 + 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0556 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.10239 - 1.04262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10239 - 1.04262i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (44.8 - 13.3i)T \) |
| good | 2 | \( 1 + (-11.7 - 4.27i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-58.4 + 331. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (23.9 - 20.0i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (823. + 4.67e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (1.31e3 - 479. i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (8.97e3 - 1.55e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (291. + 505. i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (7.01e4 + 5.88e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.65e5 - 6.01e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-2.67e4 - 2.24e4i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (1.47e5 - 2.54e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.65e5 + 2.78e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.17e5 + 6.67e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-7.17e5 + 6.01e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 4.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.82e5 + 1.60e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.50e6 + 1.26e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-7.28e5 + 2.65e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (2.73e6 - 4.74e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.49e6 + 4.32e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.31e6 + 4.76e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-6.81e5 - 2.48e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-3.25e6 - 5.63e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (6.35e5 + 3.60e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
| 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
−15.68337634980881267667447407495, −14.08703649542009734343318680890, −12.89208245837567053248834816761, −12.09489249492118378956724953473, −10.36501040744522660473101381794, −8.772608144548618701045068770927, −6.34385941192777681910957983019, −5.30983315894719690512281056436, −4.16686842996619336009922218337, −0.62062810570522405604153960421,
2.44863784368987831117501802440, 4.45007680264351471379450021714, 5.97425365647531615876145038662, 7.36277932733548055698607249012, 10.02196255650888256228708388306, 11.29436008744418922024724868505, 12.24414604511472936163576071220, 13.44780938112560104661030519957, 14.53696780147909026850689671675, 15.82345935792329989613229374784
