| L(s) = 1 | + (0.240 + 0.301i)2-s + (8.08 − 10.1i)3-s + (7.08 − 31.0i)4-s + (33.3 − 16.0i)5-s + 4.99·6-s − 195.·7-s + (22.1 − 10.6i)8-s + (16.6 + 73.0i)9-s + (12.8 + 6.18i)10-s + (−118. − 519. i)11-s + (−257. − 322. i)12-s + (330. − 159. i)13-s + (−47.0 − 58.9i)14-s + (106. − 467. i)15-s + (−909. − 438. i)16-s + (1.70e3 + 820. i)17-s + ⋯ |
| L(s) = 1 | + (0.0424 + 0.0532i)2-s + (0.518 − 0.650i)3-s + (0.221 − 0.970i)4-s + (0.595 − 0.287i)5-s + 0.0566·6-s − 1.50·7-s + (0.122 − 0.0590i)8-s + (0.0686 + 0.300i)9-s + (0.0406 + 0.0195i)10-s + (−0.295 − 1.29i)11-s + (−0.516 − 0.647i)12-s + (0.542 − 0.261i)13-s + (−0.0641 − 0.0804i)14-s + (0.122 − 0.536i)15-s + (−0.888 − 0.427i)16-s + (1.43 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.18526 - 1.45772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18526 - 1.45772i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-653. + 1.21e4i)T \) |
| good | 2 | \( 1 + (-0.240 - 0.301i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-8.08 + 10.1i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-33.3 + 16.0i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (118. + 519. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-330. + 159. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-1.70e3 - 820. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-168. + 739. i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (707. + 3.10e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-1.60e3 - 2.01e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-4.48e3 - 5.62e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 - 7.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.51e3 - 9.42e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (1.11e3 - 4.89e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (2.51e4 + 1.20e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-1.88e4 - 9.09e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-9.45e3 + 1.18e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (1.14e4 - 5.01e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-2.96e3 + 1.29e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-6.84e4 + 3.29e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + 2.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (5.73e4 - 7.19e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (5.23e4 - 6.55e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (3.09e4 + 1.35e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.30838757872890641305027944446, −13.50979998970655078082994403579, −12.68297773809291382677065349396, −10.72858208693047643455298782938, −9.757309905200915526692735304443, −8.369390895037087765349319536035, −6.58085258419164027971793773598, −5.63684727866959483080220772349, −2.91909188850695971205366131144, −1.01398965176630823911883389893,
2.77837291398822322724826259746, 3.92385124095319626117342050270, 6.29311192099763805816525832282, 7.68244175995197404973170847779, 9.545924024687888374443061684264, 9.899863582735213567840538072678, 11.95135603160399167482988602607, 12.91213411187452602871955834149, 14.05233520326614030352500437168, 15.52308977414632195005161577732
