| L(s) = 1 | + (3.28 − 6.81i)2-s + (−72.1 + 105. i)3-s + (123. + 155. i)4-s + (428. + 462. i)5-s + (484. + 839. i)6-s + (2.43e3 + 1.40e3i)7-s + (3.35e3 − 765. i)8-s + (−3.60e3 − 9.17e3i)9-s + (4.55e3 − 1.40e3i)10-s + (−4.30e3 + 5.39e3i)11-s + (−2.53e4 + 1.90e3i)12-s + (3.80e4 + 1.17e4i)13-s + (1.75e4 − 1.19e4i)14-s + (−7.98e4 + 1.20e4i)15-s + (−5.53e3 + 2.42e4i)16-s + (1.90e4 + 1.76e4i)17-s + ⋯ |
| L(s) = 1 | + (0.205 − 0.425i)2-s + (−0.891 + 1.30i)3-s + (0.484 + 0.607i)4-s + (0.686 + 0.739i)5-s + (0.373 + 0.647i)6-s + (1.01 + 0.585i)7-s + (0.818 − 0.186i)8-s + (−0.548 − 1.39i)9-s + (0.455 − 0.140i)10-s + (−0.293 + 0.368i)11-s + (−1.22 + 0.0917i)12-s + (1.33 + 0.411i)13-s + (0.457 − 0.311i)14-s + (−1.57 + 0.237i)15-s + (−0.0844 + 0.369i)16-s + (0.227 + 0.211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.13988 + 1.95557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13988 + 1.95557i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.18e6 + 3.20e6i)T \) |
| good | 2 | \( 1 + (-3.28 + 6.81i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (72.1 - 105. i)T + (-2.39e3 - 6.10e3i)T^{2} \) |
| 5 | \( 1 + (-428. - 462. i)T + (-2.91e4 + 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-2.43e3 - 1.40e3i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (4.30e3 - 5.39e3i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-3.80e4 - 1.17e4i)T + (6.73e8 + 4.59e8i)T^{2} \) |
| 17 | \( 1 + (-1.90e4 - 1.76e4i)T + (5.21e8 + 6.95e9i)T^{2} \) |
| 19 | \( 1 + (1.50e5 + 5.89e4i)T + (1.24e10 + 1.15e10i)T^{2} \) |
| 23 | \( 1 + (-3.81e5 - 5.75e4i)T + (7.48e10 + 2.30e10i)T^{2} \) |
| 29 | \( 1 + (5.11e5 + 7.50e5i)T + (-1.82e11 + 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-1.41e4 - 1.88e5i)T + (-8.43e11 + 1.27e11i)T^{2} \) |
| 37 | \( 1 + (8.13e5 - 4.69e5i)T + (1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.33e6 + 1.60e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (7.40e5 + 9.28e5i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-8.59e6 + 2.65e6i)T + (5.14e13 - 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-3.48e6 + 1.52e7i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-2.04e7 - 1.52e6i)T + (1.89e14 + 2.85e13i)T^{2} \) |
| 67 | \( 1 + (6.78e6 - 1.72e7i)T + (-2.97e14 - 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-7.73e4 - 5.13e5i)T + (-6.17e14 + 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-1.28e7 + 4.15e7i)T + (-6.66e14 - 4.54e14i)T^{2} \) |
| 79 | \( 1 + (2.90e7 - 5.03e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.01e7 - 1.37e7i)T + (8.22e14 + 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-2.26e7 + 3.32e7i)T + (-1.43e15 - 3.66e15i)T^{2} \) |
| 97 | \( 1 + (2.63e7 - 3.30e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−14.90490605670478022765971242129, −13.31557422776735359706740619859, −11.71614568428548371133551614535, −11.02428434513170291122878285209, −10.28762258452766775126558616022, −8.619253186439331848764822393209, −6.60297030900272623805681380154, −5.19790962223641717907922826730, −3.81431142647157356340433446104, −2.08674381935441133263274840206,
0.994512437769065531736735260078, 1.61656823069891518116012622391, 5.10141196503345357653237971420, 5.94963829111336880380415214890, 7.10667846550807617164370218087, 8.404448593220756293973063313503, 10.65417419780419897476144676708, 11.34286836816555207118482182551, 12.93967657641111249403265113727, 13.56014664961479075690355087693
