| L(s) = 1 | + (−12.3 − 25.7i)2-s + (27.0 − 56.1i)3-s + (−348. + 437. i)4-s + (1.12e3 + 257. i)5-s − 1.77e3·6-s − 11.8i·7-s + (8.45e3 + 1.93e3i)8-s + (1.66e3 + 2.09e3i)9-s + (−7.36e3 − 3.22e4i)10-s + (−1.25e4 − 1.56e4i)11-s + (1.51e4 + 3.14e4i)12-s + (6.78e3 − 2.97e4i)13-s + (−305. + 147. i)14-s + (4.50e4 − 5.64e4i)15-s + (−2.32e4 − 1.01e5i)16-s + (−2.10e4 − 9.24e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.774 − 1.60i)2-s + (0.333 − 0.693i)3-s + (−1.36 + 1.70i)4-s + (1.80 + 0.412i)5-s − 1.37·6-s − 0.00494i·7-s + (2.06 + 0.471i)8-s + (0.254 + 0.318i)9-s + (−0.736 − 3.22i)10-s + (−0.854 − 1.07i)11-s + (0.729 + 1.51i)12-s + (0.237 − 1.04i)13-s + (−0.00795 + 0.00382i)14-s + (0.889 − 1.11i)15-s + (−0.354 − 1.55i)16-s + (−0.252 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0793i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.996 + 0.0793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0654930 - 1.64868i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0654930 - 1.64868i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.80e6 + 1.95e6i)T \) |
| good | 2 | \( 1 + (12.3 + 25.7i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-27.0 + 56.1i)T + (-4.09e3 - 5.12e3i)T^{2} \) |
| 5 | \( 1 + (-1.12e3 - 257. i)T + (3.51e5 + 1.69e5i)T^{2} \) |
| 7 | \( 1 + 11.8iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.25e4 + 1.56e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-6.78e3 + 2.97e4i)T + (-7.34e8 - 3.53e8i)T^{2} \) |
| 17 | \( 1 + (2.10e4 + 9.24e4i)T + (-6.28e9 + 3.02e9i)T^{2} \) |
| 19 | \( 1 + (-1.16e5 - 9.27e4i)T + (3.77e9 + 1.65e10i)T^{2} \) |
| 23 | \( 1 + (1.85e5 + 2.32e5i)T + (-1.74e10 + 7.63e10i)T^{2} \) |
| 29 | \( 1 + (2.75e5 + 5.72e5i)T + (-3.11e11 + 3.91e11i)T^{2} \) |
| 31 | \( 1 + (7.57e5 - 3.64e5i)T + (5.31e11 - 6.66e11i)T^{2} \) |
| 37 | \( 1 + 1.84e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (5.17e5 - 2.49e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (1.00e6 - 1.26e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-1.66e6 - 7.31e6i)T + (-5.60e13 + 2.70e13i)T^{2} \) |
| 59 | \( 1 + (2.18e6 + 9.57e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (6.88e6 - 1.42e7i)T + (-1.19e14 - 1.49e14i)T^{2} \) |
| 67 | \( 1 + (5.62e5 - 7.05e5i)T + (-9.03e13 - 3.95e14i)T^{2} \) |
| 71 | \( 1 + (2.82e7 + 2.25e7i)T + (1.43e14 + 6.29e14i)T^{2} \) |
| 73 | \( 1 + (-4.78e6 - 1.09e6i)T + (7.26e14 + 3.49e14i)T^{2} \) |
| 79 | \( 1 - 1.32e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.95e7 - 9.41e6i)T + (1.40e15 + 1.76e15i)T^{2} \) |
| 89 | \( 1 + (4.65e7 - 9.67e7i)T + (-2.45e15 - 3.07e15i)T^{2} \) |
| 97 | \( 1 + (-9.37e7 - 1.17e8i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.47189659036732705068896861326, −12.51061364900012156466210946280, −10.81250619001077669462351150247, −10.20557395063485941521886929985, −9.046630351988589522258018312864, −7.71446246307357088222233490294, −5.65076613324317972007770166695, −2.93064149240540635194158358675, −2.13682222746744786808723463412, −0.821138719735197794841362264864,
1.63160448791263718050404298594, 4.75919398813670606694460196733, 5.88416382676851937068823148446, 7.10326392829266773935755242856, 8.840838826924608659495265084747, 9.552899737574910158839864534578, 10.21404934092741245419375377328, 13.02725057805433683609054942221, 14.06108965592417499824929863174, 15.02326179148172977854114643392
