| L(s) = 1 | + 41.2i·2-s + (−198. − 114. i)3-s − 680.·4-s + (422. + 244. i)5-s + (4.72e3 − 8.18e3i)6-s + (5.42e3 − 3.12e3i)7-s + 1.41e4i·8-s + (−3.30e3 − 5.72e3i)9-s + (−1.00e4 + 1.74e4i)10-s + 1.24e5·11-s + (1.34e5 + 7.78e4i)12-s + (2.55e5 + 4.42e5i)13-s + (1.29e5 + 2.23e5i)14-s + (−5.58e4 − 9.67e4i)15-s − 1.28e6·16-s + (5.87e5 + 1.01e6i)17-s + ⋯ |
| L(s) = 1 | + 1.29i·2-s + (−0.816 − 0.471i)3-s − 0.664·4-s + (0.135 + 0.0780i)5-s + (0.607 − 1.05i)6-s + (0.322 − 0.186i)7-s + 0.433i·8-s + (−0.0559 − 0.0969i)9-s + (−0.100 + 0.174i)10-s + 0.770·11-s + (0.542 + 0.312i)12-s + (0.687 + 1.19i)13-s + (0.240 + 0.416i)14-s + (−0.0735 − 0.127i)15-s − 1.22·16-s + (0.413 + 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.160846 - 0.764116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160846 - 0.764116i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.06e7 - 1.28e8i)T \) |
| good | 2 | \( 1 - 41.2iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (198. + 114. i)T + (2.95e4 + 5.11e4i)T^{2} \) |
| 5 | \( 1 + (-422. - 244. i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-5.42e3 + 3.12e3i)T + (1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 - 1.24e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (-2.55e5 - 4.42e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + (-5.87e5 - 1.01e6i)T + (-1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (2.76e6 + 1.59e6i)T + (3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (4.59e6 - 7.96e6i)T + (-2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (5.50e6 - 3.17e6i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-2.72e7 + 4.71e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + (2.51e7 + 1.45e7i)T + (2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + 1.18e8T + 1.34e16T^{2} \) |
| 47 | \( 1 + 1.76e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (3.34e8 - 5.79e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + 9.34e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (7.82e8 - 4.51e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (6.25e8 - 1.08e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-5.58e8 + 3.22e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.34e9 + 7.74e8i)T + (2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (3.38e8 + 5.87e8i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (1.70e9 - 2.95e9i)T + (-7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-3.99e9 - 2.30e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + 4.35e9T + 7.37e19T^{2} \) |
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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
−14.53909081913616073505068414485, −13.57674250051321682542409362444, −11.94095490387595598075681530609, −11.11870550294076552440253189560, −9.171106448682595766888543852684, −7.84908492609820987190911069965, −6.45693141288003511171597517776, −6.05559816344827740152796527560, −4.30878499028865309354616036571, −1.60569229103611182827188491894,
0.27918073497552652960795056083, 1.73141797136489226784625970295, 3.39295666415239507128357572830, 4.85166493061846842076504611768, 6.29390033423485941145342970301, 8.432256317895312908437081153156, 10.02740876607833450855785563810, 10.74371549580913476916711693299, 11.72281138336442395582511139294, 12.58682158768552974628698480222
