| L(s) = 1 | + (0.191 − 0.152i)2-s + (−5.22 + 2.04i)3-s + (−0.876 + 3.84i)4-s + (3.10 + 0.232i)5-s + (−0.687 + 1.19i)6-s + (−6.38 + 3.68i)7-s + (0.844 + 1.75i)8-s + (16.4 − 15.2i)9-s + (0.631 − 0.430i)10-s + (−0.0771 − 0.337i)11-s + (−3.29 − 21.8i)12-s + (9.78 + 6.66i)13-s + (−0.660 + 1.68i)14-s + (−16.7 + 5.15i)15-s + (−13.7 − 6.63i)16-s + (2.21 + 29.4i)17-s + ⋯ |
| L(s) = 1 | + (0.0957 − 0.0763i)2-s + (−1.74 + 0.683i)3-s + (−0.219 + 0.960i)4-s + (0.621 + 0.0465i)5-s + (−0.114 + 0.198i)6-s + (−0.912 + 0.526i)7-s + (0.105 + 0.219i)8-s + (1.83 − 1.69i)9-s + (0.0631 − 0.0430i)10-s + (−0.00701 − 0.0307i)11-s + (−0.274 − 1.82i)12-s + (0.752 + 0.512i)13-s + (−0.0471 + 0.120i)14-s + (−1.11 + 0.343i)15-s + (−0.860 − 0.414i)16-s + (0.130 + 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.333429 + 0.524627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333429 + 0.524627i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-23.4 + 36.0i)T \) |
| good | 2 | \( 1 + (-0.191 + 0.152i)T + (0.890 - 3.89i)T^{2} \) |
| 3 | \( 1 + (5.22 - 2.04i)T + (6.59 - 6.12i)T^{2} \) |
| 5 | \( 1 + (-3.10 - 0.232i)T + (24.7 + 3.72i)T^{2} \) |
| 7 | \( 1 + (6.38 - 3.68i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.0771 + 0.337i)T + (-109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-9.78 - 6.66i)T + (61.7 + 157. i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 29.4i)T + (-285. + 43.0i)T^{2} \) |
| 19 | \( 1 + (-9.02 + 9.72i)T + (-26.9 - 359. i)T^{2} \) |
| 23 | \( 1 + (-20.0 - 6.19i)T + (437. + 297. i)T^{2} \) |
| 29 | \( 1 + (2.13 + 0.837i)T + (616. + 572. i)T^{2} \) |
| 31 | \( 1 + (-30.9 + 4.66i)T + (918. - 283. i)T^{2} \) |
| 37 | \( 1 + (30.7 + 17.7i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (4.36 + 5.47i)T + (-374. + 1.63e3i)T^{2} \) |
| 47 | \( 1 + (3.79 - 16.6i)T + (-1.99e3 - 958. i)T^{2} \) |
| 53 | \( 1 + (-61.9 + 42.2i)T + (1.02e3 - 2.61e3i)T^{2} \) |
| 59 | \( 1 + (-84.5 - 40.7i)T + (2.17e3 + 2.72e3i)T^{2} \) |
| 61 | \( 1 + (1.58 - 10.4i)T + (-3.55e3 - 1.09e3i)T^{2} \) |
| 67 | \( 1 + (55.0 + 51.1i)T + (335. + 4.47e3i)T^{2} \) |
| 71 | \( 1 + (-23.8 - 77.1i)T + (-4.16e3 + 2.83e3i)T^{2} \) |
| 73 | \( 1 + (5.56 - 8.16i)T + (-1.94e3 - 4.96e3i)T^{2} \) |
| 79 | \( 1 + (-38.1 - 66.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (24.6 + 62.7i)T + (-5.04e3 + 4.68e3i)T^{2} \) |
| 89 | \( 1 + (-43.7 + 17.1i)T + (5.80e3 - 5.38e3i)T^{2} \) |
| 97 | \( 1 + (-6.15 - 26.9i)T + (-8.47e3 + 4.08e3i)T^{2} \) |
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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
−16.29786780816097054815703317558, −15.46981066068530968773643989262, −13.33988331458327814728825234883, −12.43352316801190019369105956926, −11.45711816524464378269775639239, −10.23712519555164389646646958267, −9.016470877949763606539411280837, −6.66461217500664756538921254539, −5.60410826800749632511116327147, −3.89991781699225384672483694079,
0.860851297673627058731800231281, 5.08787892890189444629466133304, 6.07885173355044142150830505066, 7.03449552978259299894877125070, 9.731580779231101975597885134545, 10.55710732997696777803141861851, 11.74819368415641867651246086462, 13.16375840388347036693839989097, 13.74473190315532496880613810018, 15.76157731121060694499340461377
