| 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
−15.57875221609093651162811474011, −14.57709067564603189535533962630, −12.93704219032091650396107769967, −12.30558743277904644882777578841, −10.29735299402809796500841438289, −9.420810685974341264753300475837, −7.39058963877998896298173167523, −6.78827019965939389219652208372, −5.61593947043516028217436237384, −2.18499721728337221681604694103,
0.33309857798251195148498149838, 2.38342315775958372434064154666, 5.01315533510921605338820346241, 6.06199845954448854593366506624, 8.896747970672868754084956597973, 9.635466358546246792319618589241, 10.61087260234908912614485269338, 11.49690124781289139931030278300, 13.05268660744134206240736760561, 14.25698240908977500341671369448
