L(s) = 1 | + (−9.62 − 3.50i)2-s + (31.7 − 34.3i)3-s + (−17.6 − 14.7i)4-s + (40.7 − 230. i)5-s + (−425. + 219. i)6-s + (500. − 420. i)7-s + (773. + 1.34e3i)8-s + (−174. − 2.18e3i)9-s + (−1.20e3 + 2.08e3i)10-s + (−328. − 1.86e3i)11-s + (−1.06e3 + 136. i)12-s + (−1.36e4 + 4.95e3i)13-s + (−6.29e3 + 2.29e3i)14-s + (−6.64e3 − 8.72e3i)15-s + (−2.24e3 − 1.27e4i)16-s + (−7.31e3 + 1.26e4i)17-s + ⋯ |
L(s) = 1 | + (−0.851 − 0.309i)2-s + (0.678 − 0.734i)3-s + (−0.137 − 0.115i)4-s + (0.145 − 0.826i)5-s + (−0.804 + 0.415i)6-s + (0.551 − 0.462i)7-s + (0.534 + 0.925i)8-s + (−0.0798 − 0.996i)9-s + (−0.379 + 0.658i)10-s + (−0.0743 − 0.421i)11-s + (−0.178 + 0.0228i)12-s + (−1.71 + 0.625i)13-s + (−0.612 + 0.223i)14-s + (−0.508 − 0.667i)15-s + (−0.136 − 0.775i)16-s + (−0.360 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.0433900 - 0.954579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0433900 - 0.954579i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-31.7 + 34.3i)T \) |
good | 2 | \( 1 + (9.62 + 3.50i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-40.7 + 230. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-500. + 420. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (328. + 1.86e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (1.36e4 - 4.95e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (7.31e3 - 1.26e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.16e4 + 3.75e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.19e4 - 5.19e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-8.52e4 - 3.10e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.14e5 + 9.62e4i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (-4.02e4 + 6.96e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.81e5 + 6.59e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (8.69e4 + 4.93e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-6.15e5 + 5.16e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + 1.14e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.99e5 + 1.69e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.71e6 + 1.43e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (1.17e5 - 4.27e4i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-8.50e5 + 1.47e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (8.68e5 + 1.50e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.67e6 - 9.75e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.05e6 - 7.47e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (5.87e6 + 1.01e7i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.94e6 - 1.10e7i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−14.92408113280258500739233692009, −13.84752048101610556497529142583, −12.71216861250724350108842849990, −11.13280956229582325580695876924, −9.436552855913784060239254115167, −8.621535463836154252477335056271, −7.28974898611688636767580099960, −4.83574954286596179137357011947, −2.02846352674634475378219579455, −0.58835528981639288532538413865,
2.64297083877974523366848038744, 4.70254172946964401352457827369, 7.24987587903969382656664569385, 8.416160498839489867093074403395, 9.704608860814782355555457961292, 10.60597679863343949978008954116, 12.63124397023910487904320101766, 14.41096362411193846155359934178, 15.03971731963161736101800944497, 16.46674064899674352761782505620