| L(s) = 1 | + (−1.34 + 0.775i)2-s + (−126. + 219. i)4-s + (604. + 349. i)5-s + (−1.12e3 − 1.94e3i)7-s − 790. i·8-s − 1.08e3·10-s + (−1.88e4 + 1.08e4i)11-s + (−9.20e3 + 1.59e4i)13-s + (3.02e3 + 1.74e3i)14-s + (−3.18e4 − 5.51e4i)16-s + 5.65e4i·17-s − 2.12e5·19-s + (−1.53e5 + 8.85e4i)20-s + (1.68e4 − 2.91e4i)22-s + (−1.43e4 − 8.31e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.0839 + 0.0484i)2-s + (−0.495 + 0.857i)4-s + (0.967 + 0.558i)5-s + (−0.468 − 0.811i)7-s − 0.192i·8-s − 0.108·10-s + (−1.28 + 0.742i)11-s + (−0.322 + 0.558i)13-s + (0.0786 + 0.0453i)14-s + (−0.485 − 0.841i)16-s + 0.676i·17-s − 1.62·19-s + (−0.958 + 0.553i)20-s + (0.0719 − 0.124i)22-s + (−0.0514 − 0.0297i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.106319 + 0.711969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106319 + 0.711969i\) |
| \(L(5)\) |
|
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 | \( 1 + (1.34 - 0.775i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (-604. - 349. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.12e3 + 1.94e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.88e4 - 1.08e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (9.20e3 - 1.59e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 5.65e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.12e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (1.43e4 + 8.31e3i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-2.94e5 + 1.70e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.29e4 - 1.43e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.11e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.60e6 - 2.08e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.28e6 - 2.23e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (8.10e6 - 4.67e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 4.75e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-8.39e6 - 4.84e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.04e6 + 5.27e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.02e7 - 1.77e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.08e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 9.02e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.67e7 - 2.89e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-5.09e7 + 2.94e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 8.65e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.68e7 + 8.12e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−16.34026149381173917402842163956, −14.65885450897190842734006854191, −13.37988949205536700232121093562, −12.68628688138911746671827738096, −10.62519575435641360496034460666, −9.624837981019068466009552177383, −7.904313625800402241080204690624, −6.51661922682213965542142715039, −4.38241475523637116639026395729, −2.49918899876709228862944098418,
0.31540587296957701323622158260, 2.32955285927186229820263554149, 5.13706473478957868762148449134, 6.00749699062756338253824886398, 8.529511380384173850581566774319, 9.615712534749767948964793262186, 10.71550348849380647637507266349, 12.75729276136444990449432719109, 13.54012515380190388668465104679, 14.93256648665605225490806603590
