| L(s) = 1 | + (22.1 − 12.7i)2-s + (199. − 344. i)4-s + (−570. − 329. i)5-s + (−1.51e3 − 2.61e3i)7-s − 3.63e3i·8-s − 1.68e4·10-s + (8.01e3 − 4.62e3i)11-s + (2.17e4 − 3.77e4i)13-s + (−6.70e4 − 3.86e4i)14-s + (4.44e3 + 7.70e3i)16-s + 1.07e5i·17-s − 2.23e4·19-s + (−2.27e5 + 1.31e5i)20-s + (1.18e5 − 2.05e5i)22-s + (1.15e5 + 6.69e4i)23-s + ⋯ |
| L(s) = 1 | + (1.38 − 0.799i)2-s + (0.777 − 1.34i)4-s + (−0.912 − 0.526i)5-s + (−0.629 − 1.09i)7-s − 0.888i·8-s − 1.68·10-s + (0.547 − 0.316i)11-s + (0.762 − 1.32i)13-s + (−1.74 − 1.00i)14-s + (0.0678 + 0.117i)16-s + 1.28i·17-s − 0.171·19-s + (−1.41 + 0.819i)20-s + (0.505 − 0.875i)22-s + (0.414 + 0.239i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.02577 - 2.66111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02577 - 2.66111i\) |
| \(L(5)\) |
|
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 | \( 1 + (-22.1 + 12.7i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (570. + 329. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.51e3 + 2.61e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-8.01e3 + 4.62e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-2.17e4 + 3.77e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.07e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.23e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.15e5 - 6.69e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-2.19e5 + 1.26e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-3.93e5 + 6.82e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.55e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.91e6 - 1.10e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (2.77e5 + 4.79e5i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.05e6 - 1.18e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + 7.83e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (4.48e6 + 2.59e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.77e6 + 8.26e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (4.31e6 - 7.47e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.31e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-4.50e6 - 7.80e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.31e7 + 1.91e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 1.41e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-7.52e7 - 1.30e8i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−14.78210702677580791373249314997, −13.34896785573354135793159453996, −12.74669352574999468866147954308, −11.42272680957271593120361159372, −10.36549146384626025627229796742, −8.126336412398584329796629188789, −6.10236487423138555974725882432, −4.26316381974166204523318816687, −3.41105036603518156542593345078, −0.890815246120218763470979686393,
3.05965531414419160474022603974, 4.46302770526400839678335504908, 6.19850837451422248330409491323, 7.19882100144080750760206453190, 9.108951151818190139156767200504, 11.52548274245976527488481597492, 12.30149548862555911686897296181, 13.73995118678843116799744475538, 14.81083647076785556102962363676, 15.72275617634463188680486550833
