| L(s) = 1 | + (−0.245 − 0.308i)2-s + (4.65 − 5.84i)3-s + (7.08 − 31.0i)4-s + (−95.9 + 46.2i)5-s − 2.94·6-s − 56.9·7-s + (−22.6 + 10.9i)8-s + (41.6 + 182. i)9-s + (37.8 + 18.2i)10-s + (15.4 + 67.7i)11-s + (−148. − 186. i)12-s + (−421. + 202. i)13-s + (13.9 + 17.5i)14-s + (−177. + 776. i)15-s + (−909. − 437. i)16-s + (−1.78e3 − 857. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0434 − 0.0544i)2-s + (0.298 − 0.374i)3-s + (0.221 − 0.970i)4-s + (−1.71 + 0.826i)5-s − 0.0334·6-s − 0.439·7-s + (−0.125 + 0.0603i)8-s + (0.171 + 0.750i)9-s + (0.119 + 0.0576i)10-s + (0.0385 + 0.168i)11-s + (−0.297 − 0.373i)12-s + (−0.691 + 0.332i)13-s + (0.0190 + 0.0239i)14-s + (−0.203 + 0.890i)15-s + (−0.887 − 0.427i)16-s + (−1.49 − 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0109471 + 0.0588478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0109471 + 0.0588478i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.18e4 - 2.60e3i)T \) |
| good | 2 | \( 1 + (0.245 + 0.308i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-4.65 + 5.84i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (95.9 - 46.2i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 56.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-15.4 - 67.7i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (421. - 202. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (1.78e3 + 857. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-371. + 1.62e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-582. - 2.55e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (1.91e3 + 2.39e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (3.75e3 + 4.71e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 + 1.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-5.21e3 - 6.54e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (1.51e3 - 6.63e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (2.91e4 + 1.40e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-3.27e4 - 1.57e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.67e4 - 2.09e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-2.71e3 + 1.19e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-7.11e3 + 3.11e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (2.61e4 - 1.25e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 823.T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.87e4 - 2.35e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-3.15e4 + 3.95e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (1.28e4 + 5.63e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.50114693556452597744302956311, −13.22903849285861742785337269668, −11.53164776409014090797504620635, −10.97371935579524297241183511829, −9.375145830390012187344801151345, −7.57537856370424728846312198718, −6.82530329430166155900720438599, −4.58474194141983013202491367631, −2.57637885440120963855739506474, −0.03025168188454925319640351764,
3.41494396496293600544209717819, 4.34172949925564732244116471355, 6.97442826572639823218747010066, 8.223595761617119944868856048044, 9.047028634576154560941874857791, 11.03534696483293239701200270343, 12.41771770775482494424618756521, 12.63041866587381246250194795454, 14.82314547744100289625243001779, 15.80024569627695962203101684324
