| L(s) = 1 | + (−9.86 + 12.3i)2-s + (−10.7 + 1.61i)3-s + (−27.2 − 119. i)4-s + (180. − 122. i)5-s + (85.6 − 148. i)6-s + (797. + 1.38e3i)7-s + (−79.5 − 38.2i)8-s + (−1.97e3 + 610. i)9-s + (−258. + 3.44e3i)10-s + (−774. + 3.39e3i)11-s + (484. + 1.23e3i)12-s + (−123. − 1.65e3i)13-s + (−2.49e4 − 3.76e3i)14-s + (−1.73e3 + 1.60e3i)15-s + (1.53e4 − 7.40e3i)16-s + (1.62e4 + 1.10e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.872 + 1.09i)2-s + (−0.228 + 0.0345i)3-s + (−0.212 − 0.932i)4-s + (0.645 − 0.439i)5-s + (0.161 − 0.280i)6-s + (0.879 + 1.52i)7-s + (−0.0549 − 0.0264i)8-s + (−0.904 + 0.278i)9-s + (−0.0816 + 1.08i)10-s + (−0.175 + 0.768i)11-s + (0.0809 + 0.206i)12-s + (−0.0156 − 0.208i)13-s + (−2.43 − 0.366i)14-s + (−0.132 + 0.123i)15-s + (0.938 − 0.451i)16-s + (0.802 + 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.196655 - 0.507886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196655 - 0.507886i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.23e5 - 4.08e5i)T \) |
| good | 2 | \( 1 + (9.86 - 12.3i)T + (-28.4 - 124. i)T^{2} \) |
| 3 | \( 1 + (10.7 - 1.61i)T + (2.08e3 - 644. i)T^{2} \) |
| 5 | \( 1 + (-180. + 122. i)T + (2.85e4 - 7.27e4i)T^{2} \) |
| 7 | \( 1 + (-797. - 1.38e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (774. - 3.39e3i)T + (-1.75e7 - 8.45e6i)T^{2} \) |
| 13 | \( 1 + (123. + 1.65e3i)T + (-6.20e7 + 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-1.62e4 - 1.10e4i)T + (1.49e8 + 3.81e8i)T^{2} \) |
| 19 | \( 1 + (2.21e4 + 6.81e3i)T + (7.38e8 + 5.03e8i)T^{2} \) |
| 23 | \( 1 + (5.43e4 + 5.04e4i)T + (2.54e8 + 3.39e9i)T^{2} \) |
| 29 | \( 1 + (1.92e5 + 2.89e4i)T + (1.64e10 + 5.08e9i)T^{2} \) |
| 31 | \( 1 + (9.02e4 + 2.30e5i)T + (-2.01e10 + 1.87e10i)T^{2} \) |
| 37 | \( 1 + (1.68e5 - 2.91e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (2.07e5 - 2.60e5i)T + (-4.33e10 - 1.89e11i)T^{2} \) |
| 47 | \( 1 + (1.05e5 + 4.63e5i)T + (-4.56e11 + 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-1.38e5 + 1.85e6i)T + (-1.16e12 - 1.75e11i)T^{2} \) |
| 59 | \( 1 + (-4.11e5 + 1.98e5i)T + (1.55e12 - 1.94e12i)T^{2} \) |
| 61 | \( 1 + (4.54e5 - 1.15e6i)T + (-2.30e12 - 2.13e12i)T^{2} \) |
| 67 | \( 1 + (2.01e6 + 6.22e5i)T + (5.00e12 + 3.41e12i)T^{2} \) |
| 71 | \( 1 + (1.14e6 - 1.06e6i)T + (6.79e11 - 9.06e12i)T^{2} \) |
| 73 | \( 1 + (-4.23e5 - 5.64e6i)T + (-1.09e13 + 1.64e12i)T^{2} \) |
| 79 | \( 1 + (-2.60e6 - 4.51e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (4.87e6 - 7.35e5i)T + (2.59e13 - 7.99e12i)T^{2} \) |
| 89 | \( 1 + (-1.08e7 + 1.63e6i)T + (4.22e13 - 1.30e13i)T^{2} \) |
| 97 | \( 1 + (-2.71e5 + 1.18e6i)T + (-7.27e13 - 3.50e13i)T^{2} \) |
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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
−15.14091401766533949462407708774, −14.66809613383004309986494917670, −12.76790349526225549402779919841, −11.58926066298391566516557329544, −9.834982419420634104762891978125, −8.712082135035681707997389384282, −7.952970462032991196312843567774, −6.02674966728915789147661263564, −5.30948659649610856646916557787, −2.07864268281419510112254773437,
0.30155456506777471712669598924, 1.72135155805269593212373944218, 3.48199102173636645566001738889, 5.76092044500818301723307364654, 7.63438799443331079810058605739, 9.024344320873288041966754378377, 10.44647010498998242836144314354, 10.91230991128153986218562739799, 12.03071213678320251832097367386, 13.82936176385351315535769115952
