| L(s) = 1 | + (17.6 + 6.40i)2-s + (−7.16 + 46.2i)3-s + (170. + 143. i)4-s + (−14.0 + 79.8i)5-s + (−422. + 767. i)6-s + (245. − 206. i)7-s + (889. + 1.54e3i)8-s + (−2.08e3 − 662. i)9-s + (−759. + 1.31e3i)10-s + (695. + 3.94e3i)11-s + (−7.84e3 + 6.86e3i)12-s + (3.68e3 − 1.34e3i)13-s + (5.64e3 − 2.05e3i)14-s + (−3.59e3 − 1.22e3i)15-s + (831. + 4.71e3i)16-s + (1.66e4 − 2.88e4i)17-s + ⋯ |
| L(s) = 1 | + (1.55 + 0.566i)2-s + (−0.153 + 0.988i)3-s + (1.33 + 1.11i)4-s + (−0.0503 + 0.285i)5-s + (−0.797 + 1.45i)6-s + (0.270 − 0.227i)7-s + (0.614 + 1.06i)8-s + (−0.953 − 0.302i)9-s + (−0.240 + 0.416i)10-s + (0.157 + 0.893i)11-s + (−1.31 + 1.14i)12-s + (0.465 − 0.169i)13-s + (0.550 − 0.200i)14-s + (−0.274 − 0.0935i)15-s + (0.0507 + 0.287i)16-s + (0.823 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.16773 + 2.92432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16773 + 2.92432i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.16 - 46.2i)T \) |
| good | 2 | \( 1 + (-17.6 - 6.40i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (14.0 - 79.8i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-245. + 206. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-695. - 3.94e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-3.68e3 + 1.34e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-1.66e4 + 2.88e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.26e4 - 3.92e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.85e4 + 2.39e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-4.14e4 - 1.50e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.81e5 + 1.52e5i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (-1.62e5 + 2.81e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (8.59e4 - 3.12e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (5.00e4 + 2.83e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (8.31e5 - 6.97e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + 1.18e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.64e4 + 1.49e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (3.01e5 - 2.53e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.87e5 + 1.04e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (2.52e6 - 4.37e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.37e6 + 4.11e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-5.52e6 - 2.01e6i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (1.83e6 + 6.69e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (6.02e6 + 1.04e7i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (2.26e6 + 1.28e7i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.89738275605660347057373835541, −14.65214573178093935977443604614, −14.16261722564990009463814760110, −12.47640754579752842967770064956, −11.33229401576333892951702762566, −9.743460549806342067853745525089, −7.48495722163149445034236890143, −5.80330722518277491569121819239, −4.58754899449367034655989464467, −3.26953743366658733819856059678,
1.43070321115754319908734705338, 3.24201306383978344078102922625, 5.23352525626480890122791704694, 6.42500446641408259581768082856, 8.410059180215152955639622488232, 10.98252810671669264404959107463, 11.89469204308690443238530845475, 12.93456818067867934953733960428, 13.79399145505988701128970555521, 14.83134139930331239078477234575
