| L(s) = 1 | + 5.12·2-s + (23.6 + 40.9i)3-s − 101.·4-s + (246. + 427. i)5-s + (121. + 209. i)6-s + (159. − 276. i)7-s − 1.17e3·8-s + (−26.1 + 45.2i)9-s + (1.26e3 + 2.18e3i)10-s − 4.64e3·11-s + (−2.40e3 − 4.17e3i)12-s + (−3.89e3 + 6.74e3i)13-s + (818. − 1.41e3i)14-s + (−1.16e4 + 2.02e4i)15-s + 7.00e3·16-s + (−1.19e3 + 2.07e3i)17-s + ⋯ |
| L(s) = 1 | + 0.452·2-s + (0.505 + 0.876i)3-s − 0.795·4-s + (0.883 + 1.53i)5-s + (0.228 + 0.396i)6-s + (0.176 − 0.305i)7-s − 0.812·8-s + (−0.0119 + 0.0207i)9-s + (0.399 + 0.692i)10-s − 1.05·11-s + (−0.402 − 0.696i)12-s + (−0.491 + 0.851i)13-s + (0.0797 − 0.138i)14-s + (−0.893 + 1.54i)15-s + 0.427·16-s + (−0.0591 + 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.633i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.672795 + 1.88532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.672795 + 1.88532i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.42e4 + 5.16e5i)T \) |
| good | 2 | \( 1 - 5.12T + 128T^{2} \) |
| 3 | \( 1 + (-23.6 - 40.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-246. - 427. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-159. + 276. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 4.64e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.89e3 - 6.74e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.19e3 - 2.07e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (5.63e3 + 9.76e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.71e4 - 4.69e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (8.32e4 - 1.44e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (5.62e4 + 9.74e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (5.52e4 + 9.56e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 7.91e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 8.42e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-6.23e5 - 1.07e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.35e6 + 2.34e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.59e6 - 2.75e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.01e5 + 1.75e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.70e6 - 4.68e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (6.88e5 - 1.19e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.20e6 - 2.08e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.98e6 + 6.89e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.18e5T + 8.07e13T^{2} \) |
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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
−14.60472507458937371390692719253, −14.09556416756319014635605623121, −12.97137571727337317744420915324, −10.98255058549475921340385197871, −9.978516050128863905854678177601, −9.131100812152379104455876825159, −7.15797008847874195705890188134, −5.48926689275283110504146129687, −3.94840250548748735461427277228, −2.66189654504406348841342199262,
0.69285784700381731430935264279, 2.36039940892081659600801328023, 4.77645838925508977441018300562, 5.67017839098078185313507006831, 7.949773401490637965857081438901, 8.777288997741208064565348130045, 10.02678773392775313922926368614, 12.43088493488046158111994693579, 12.97889586092946252163642370910, 13.53952777379996948890092892345
