| L(s) = 1 | + (−6.31 − 7.92i)2-s + (−18.2 + 22.9i)3-s + (−15.7 + 68.8i)4-s + (−46.4 + 22.3i)5-s + 296.·6-s − 129.·7-s + (352. − 169. i)8-s + (−136. − 599. i)9-s + (470. + 226. i)10-s + (113. + 497. i)11-s + (−1.28e3 − 1.61e3i)12-s + (−517. + 249. i)13-s + (820. + 1.02e3i)14-s + (336. − 1.47e3i)15-s + (−1.53e3 − 738. i)16-s + (−0.150 − 0.0723i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 1.40i)2-s + (−1.17 + 1.46i)3-s + (−0.491 + 2.15i)4-s + (−0.831 + 0.400i)5-s + 3.36·6-s − 1.00·7-s + (1.94 − 0.937i)8-s + (−0.563 − 2.46i)9-s + (1.48 + 0.716i)10-s + (0.283 + 1.24i)11-s + (−2.58 − 3.24i)12-s + (−0.849 + 0.409i)13-s + (1.11 + 1.40i)14-s + (0.385 − 1.69i)15-s + (−1.49 − 0.721i)16-s + (−0.000126 − 6.06e−5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0533611 - 0.0642412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0533611 - 0.0642412i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-400. + 1.21e4i)T \) |
| good | 2 | \( 1 + (6.31 + 7.92i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (18.2 - 22.9i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (46.4 - 22.3i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 129.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-113. - 497. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (517. - 249. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (0.150 + 0.0723i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-537. + 2.35e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-455. - 1.99e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (243. + 305. i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-1.15e3 - 1.45e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 + 3.84e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.00e3 - 8.77e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (4.92e3 - 2.15e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.01e4 + 4.88e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-5.91e3 - 2.85e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-1.72e4 + 2.16e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-5.59e3 + 2.45e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.21e4 + 5.33e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (4.82e4 - 2.32e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + 770.T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.48e4 + 1.85e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (6.38e4 - 8.00e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-3.89e3 - 1.70e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−15.18170762220623012690141833231, −12.54158333446161197620499243260, −11.69697696311657189888055590612, −10.95471824758351352751590742431, −9.738827411896756808902030655788, −9.400149174601422902803784697407, −7.06283182027775747982609919007, −4.52690135918366314750986538936, −3.25490754230201930250499135324, −0.12159873498622237591037820083,
0.76546937469551301238584727105, 5.58003963535850599471982022977, 6.46461219222907929142328077052, 7.55219302884240141763040134985, 8.431102914205302730026333953308, 10.26009228854365338489178122944, 11.77889042260016553948117100600, 12.84900053611715390037843383083, 14.27295909994641205287014069509, 16.05497785644496161655754723703
