| L(s) = 1 | + (0.252 + 1.10i)2-s + (−51.5 − 47.8i)3-s + (114. − 54.9i)4-s + (396. − 59.8i)5-s + (39.9 − 69.1i)6-s + (−274. − 474. i)7-s + (180. + 226. i)8-s + (205. + 2.74e3i)9-s + (166. + 424. i)10-s + (6.63e3 + 3.19e3i)11-s + (−8.51e3 − 2.62e3i)12-s + (2.29e3 − 5.84e3i)13-s + (456. − 423. i)14-s + (−2.33e4 − 1.58e4i)15-s + (9.90e3 − 1.24e4i)16-s + (−3.26e4 − 4.92e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0223 + 0.0978i)2-s + (−1.10 − 1.02i)3-s + (0.891 − 0.429i)4-s + (1.41 − 0.214i)5-s + (0.0754 − 0.130i)6-s + (−0.301 − 0.523i)7-s + (0.124 + 0.156i)8-s + (0.0941 + 1.25i)9-s + (0.0526 + 0.134i)10-s + (1.50 + 0.724i)11-s + (−1.42 − 0.438i)12-s + (0.289 − 0.738i)13-s + (0.0444 − 0.0412i)14-s + (−1.78 − 1.21i)15-s + (0.604 − 0.758i)16-s + (−1.61 − 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.15721 - 1.53805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15721 - 1.53805i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.14e5 + 8.40e4i)T \) |
| good | 2 | \( 1 + (-0.252 - 1.10i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (51.5 + 47.8i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-396. + 59.8i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (274. + 474. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-6.63e3 - 3.19e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (-2.29e3 + 5.84e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (3.26e4 + 4.92e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (-3.46e3 + 4.62e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (1.33e4 - 9.09e3i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (1.39e5 - 1.29e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (-1.62e5 - 5.00e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (5.28e4 - 9.14e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-8.26e3 - 3.62e4i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-3.03e5 + 1.46e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-1.86e5 - 4.74e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (-1.30e6 + 1.63e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (1.24e6 - 3.84e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-1.31e5 + 1.74e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (-2.30e6 - 1.57e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (8.88e5 - 2.26e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (-2.20e6 - 3.81e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-6.77e6 - 6.29e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (-3.31e6 - 3.07e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (5.28e6 + 2.54e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−13.78643721057155593712798385541, −12.95391854123010065790533894851, −11.68679345718189600268369657019, −10.70479978840002985460909015790, −9.370368364101892270314911875105, −6.80266917891147024286257703623, −6.62452725172368323112414403632, −5.25831589712445635720318080511, −2.02227358174398434655231435620, −0.942565142445736597930213618100,
1.91586056127160285122366966163, 3.95205282804935281249034273533, 6.08549022812930328288269359083, 6.27511236922305488849385411081, 9.033697332603333901029175244363, 10.11982059634342531049512589184, 11.22228403690019208888166674420, 11.99288701014542077512397816556, 13.61809482104249739219769465930, 15.02250159886336255500339962684
