| L(s) = 1 | + (1.04 + 4.57i)2-s + (18.6 + 17.2i)3-s + (95.4 − 45.9i)4-s + (−284. + 42.8i)5-s + (−59.6 + 103. i)6-s + (−811. − 1.40e3i)7-s + (684. + 858. i)8-s + (−115. − 1.53e3i)9-s + (−493. − 1.25e3i)10-s + (−3.64e3 − 1.75e3i)11-s + (2.57e3 + 794. i)12-s + (−4.27e3 + 1.08e4i)13-s + (5.58e3 − 5.18e3i)14-s + (−6.04e3 − 4.12e3i)15-s + (5.24e3 − 6.57e3i)16-s + (−7.77e3 − 1.17e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0923 + 0.404i)2-s + (0.398 + 0.369i)3-s + (0.745 − 0.359i)4-s + (−1.01 + 0.153i)5-s + (−0.112 + 0.195i)6-s + (−0.894 − 1.54i)7-s + (0.472 + 0.592i)8-s + (−0.0526 − 0.702i)9-s + (−0.156 − 0.397i)10-s + (−0.824 − 0.397i)11-s + (0.430 + 0.132i)12-s + (−0.539 + 1.37i)13-s + (0.544 − 0.504i)14-s + (−0.462 − 0.315i)15-s + (0.319 − 0.401i)16-s + (−0.383 − 0.0578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.724646 - 0.850203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.724646 - 0.850203i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.20e5 + 2.10e4i)T \) |
| good | 2 | \( 1 + (-1.04 - 4.57i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-18.6 - 17.2i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (284. - 42.8i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (811. + 1.40e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.64e3 + 1.75e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (4.27e3 - 1.08e4i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (7.77e3 + 1.17e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (-3.75e3 + 5.00e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (-7.00e4 + 4.77e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (1.49e5 - 1.38e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (9.58e3 + 2.95e3i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-1.13e5 + 1.96e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-6.60e4 - 2.89e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-5.50e5 + 2.65e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-2.46e5 - 6.26e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (1.07e5 - 1.34e5i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (5.50e5 - 1.69e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-9.86e4 + 1.31e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (-2.20e6 - 1.50e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (1.82e5 - 4.63e5i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (3.98e6 + 6.90e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (6.23e6 + 5.78e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (2.41e5 + 2.24e5i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-1.29e7 - 6.24e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−14.40723845802933491429226787112, −13.16650235463620497378375003647, −11.47026162563694740542923757502, −10.60948108524756515906424216641, −9.174051091340203636009843374631, −7.31214984018449217183094900038, −6.77650875134933912977579681000, −4.46066916453505492856295733336, −3.06619485940036703979084182244, −0.39877505282583360934327448617,
2.26251376520312595699531741632, 3.28394497261644830648082956402, 5.57467928544010248274488510967, 7.47911468754034812827337275581, 8.224249344796047583987243126400, 10.03711013263559667901489964122, 11.46724395190295869921922402272, 12.53877792136678200371317058734, 12.99433107428450617150678276049, 15.29121460998669924204351378359
