| L(s) = 1 | + (2.94 + 12.9i)2-s + (59.6 + 55.3i)3-s + (−42.6 + 20.5i)4-s + (213. − 32.1i)5-s + (−538. + 933. i)6-s + (−170. − 295. i)7-s + (665. + 834. i)8-s + (331. + 4.42e3i)9-s + (1.04e3 + 2.65e3i)10-s + (−440. − 212. i)11-s + (−3.68e3 − 1.13e3i)12-s + (1.37e3 − 3.51e3i)13-s + (3.31e3 − 3.07e3i)14-s + (1.44e4 + 9.88e3i)15-s + (−1.25e4 + 1.57e4i)16-s + (−1.65e3 − 250. i)17-s + ⋯ |
| L(s) = 1 | + (0.260 + 1.14i)2-s + (1.27 + 1.18i)3-s + (−0.333 + 0.160i)4-s + (0.762 − 0.114i)5-s + (−1.01 + 1.76i)6-s + (−0.188 − 0.325i)7-s + (0.459 + 0.576i)8-s + (0.151 + 2.02i)9-s + (0.329 + 0.839i)10-s + (−0.0997 − 0.0480i)11-s + (−0.615 − 0.189i)12-s + (0.174 − 0.443i)13-s + (0.322 − 0.299i)14-s + (1.10 + 0.755i)15-s + (−0.768 + 0.964i)16-s + (−0.0819 − 0.0123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.30144 + 3.39313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30144 + 3.39313i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.00e5 + 4.26e5i)T \) |
| good | 2 | \( 1 + (-2.94 - 12.9i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-59.6 - 55.3i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-213. + 32.1i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (170. + 295. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (440. + 212. i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (-1.37e3 + 3.51e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (1.65e3 + 250. i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (-4.45e3 + 5.94e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (4.00e4 - 2.73e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.23e5 + 1.15e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (1.80e5 + 5.56e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-3.63e4 + 6.29e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.52e4 - 3.29e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-4.17e5 + 2.01e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-6.66e5 - 1.69e6i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (7.82e5 - 9.81e5i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (1.28e6 - 3.95e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (8.11e3 - 1.08e5i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (-2.43e5 - 1.66e5i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (1.40e6 - 3.56e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (-1.76e6 - 3.05e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-3.63e6 - 3.37e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (2.57e6 + 2.38e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (8.52e6 + 4.10e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−15.18670617203920419932743245023, −13.84005316469581495760667400750, −13.53350566357681565013794807460, −10.87258085676420971234723768729, −9.702560670000461623521440196365, −8.662021586753847494739606107019, −7.37459076761923333595573770087, −5.63728314324349083270755077789, −4.30248759679322779990680487987, −2.50067177703479979709670679635,
1.48852308262063486887417186005, 2.30545073058060347615884201956, 3.58823753637751903565319758834, 6.32297337474223092908765328502, 7.74422404030543663897929291593, 9.153933351488511330188405735642, 10.32497845850242054176227619018, 12.10862725505678694716035141805, 12.71476415894271514026703381076, 13.82250551394490527029048078639
