| L(s) = 1 | + (−7.11 − 12.3i)2-s + (−37.2 + 64.5i)4-s + (145. − 251. i)5-s + (−555. − 962. i)7-s − 760.·8-s − 4.13e3·10-s + (2.24e3 + 3.88e3i)11-s + (−1.21e3 + 2.11e3i)13-s + (−7.91e3 + 1.37e4i)14-s + (1.01e4 + 1.76e4i)16-s − 1.59e4·17-s − 4.99e4·19-s + (1.08e4 + 1.87e4i)20-s + (3.19e4 − 5.53e4i)22-s + (3.46e4 − 6.00e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.628 − 1.08i)2-s + (−0.291 + 0.504i)4-s + (0.519 − 0.900i)5-s + (−0.612 − 1.06i)7-s − 0.525·8-s − 1.30·10-s + (0.508 + 0.880i)11-s + (−0.153 + 0.266i)13-s + (−0.770 + 1.33i)14-s + (0.621 + 1.07i)16-s − 0.785·17-s − 1.67·19-s + (0.302 + 0.524i)20-s + (0.639 − 1.10i)22-s + (0.594 − 1.02i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.238595 + 0.695963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238595 + 0.695963i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (7.11 + 12.3i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-145. + 251. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (555. + 962. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.24e3 - 3.88e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.21e3 - 2.11e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.46e4 + 6.00e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.70e4 + 8.14e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-9.96e3 + 1.72e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.21e5 + 2.09e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.15e5 + 7.20e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (8.00e4 + 1.38e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 3.11e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.56e5 - 2.70e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.87e4 - 4.97e4i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.05e6 + 3.55e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 4.03e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.23e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (4.89e5 + 8.47e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.85e6 + 3.20e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 2.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.75e6 + 3.03e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−14.99193550391089009840391981422, −13.25219992214391280262035327951, −12.42889510688306657398837351148, −10.85690098038268576223502375146, −9.799143300142463276684941853015, −8.809721058244128893308720777516, −6.60736417647931661965064962520, −4.26348792682242192329613420885, −1.99809531518282661039187242748, −0.43415453425917737568884053680,
2.83552683462618708942594630337, 5.94215545693162046461928432434, 6.69840620638409214891661757874, 8.477589074634955881344557682299, 9.525422250286428048794756327090, 11.21354774754741192430645383192, 12.90229641250030987712286037868, 14.56902363925864137685902785812, 15.35015339125728715739078835784, 16.51142105956200106452347282898
