| L(s) = 1 | + (−3.83 − 4.81i)2-s + (−10.8 + 13.6i)3-s + (−1.31 + 5.75i)4-s + (77.8 − 37.5i)5-s + 107.·6-s − 36.5·7-s + (−144. + 69.7i)8-s + (−13.6 − 59.9i)9-s + (−479. − 230. i)10-s + (−163. − 715. i)11-s + (−64.2 − 80.5i)12-s + (−808. + 389. i)13-s + (140. + 175. i)14-s + (−335. + 1.47e3i)15-s + (1.06e3 + 511. i)16-s + (−1.33e3 − 644. i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.850i)2-s + (−0.697 + 0.875i)3-s + (−0.0410 + 0.179i)4-s + (1.39 − 0.670i)5-s + 1.21·6-s − 0.281·7-s + (−0.799 + 0.385i)8-s + (−0.0562 − 0.246i)9-s + (−1.51 − 0.730i)10-s + (−0.406 − 1.78i)11-s + (−0.128 − 0.161i)12-s + (−1.32 + 0.639i)13-s + (0.191 + 0.239i)14-s + (−0.385 + 1.68i)15-s + (1.03 + 0.499i)16-s + (−1.12 − 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0334746 - 0.491968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0334746 - 0.491968i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.16e4 + 3.43e3i)T \) |
| good | 2 | \( 1 + (3.83 + 4.81i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (10.8 - 13.6i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-77.8 + 37.5i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 36.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + (163. + 715. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (808. - 389. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (1.33e3 + 644. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-129. + 566. i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (392. + 1.72e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-2.23e3 - 2.80e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (4.71e3 + 5.90e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 + 8.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.11e4 - 1.39e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-1.17e3 + 5.16e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.16e4 - 5.60e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-5.44e3 - 2.62e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-2.13e4 + 2.68e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (656. - 2.87e3i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.65e3 + 7.24e3i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-601. + 289. i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + 5.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.25e4 + 1.57e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.42e4 + 8.05e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (4.79e3 + 2.09e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.25698240908977500341671369448, −13.05268660744134206240736760561, −11.49690124781289139931030278300, −10.61087260234908912614485269338, −9.635466358546246792319618589241, −8.896747970672868754084956597973, −6.06199845954448854593366506624, −5.01315533510921605338820346241, −2.38342315775958372434064154666, −0.33309857798251195148498149838,
2.18499721728337221681604694103, 5.61593947043516028217436237384, 6.78827019965939389219652208372, 7.39058963877998896298173167523, 9.420810685974341264753300475837, 10.29735299402809796500841438289, 12.30558743277904644882777578841, 12.93704219032091650396107769967, 14.57709067564603189535533962630, 15.57875221609093651162811474011
