| L(s) = 1 | + (−2.90 − 12.7i)2-s + (−23.7 − 22.0i)3-s + (−38.0 + 18.3i)4-s + (272. − 41.0i)5-s + (−211. + 366. i)6-s + (−480. − 831. i)7-s + (−697. − 874. i)8-s + (−84.9 − 1.13e3i)9-s + (−1.31e3 − 3.34e3i)10-s + (−717. − 345. i)11-s + (1.30e3 + 403. i)12-s + (−1.26e3 + 3.22e3i)13-s + (−9.18e3 + 8.52e3i)14-s + (−7.37e3 − 5.02e3i)15-s + (−1.24e4 + 1.56e4i)16-s + (1.76e4 + 2.66e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.256 − 1.12i)2-s + (−0.508 − 0.471i)3-s + (−0.297 + 0.143i)4-s + (0.974 − 0.146i)5-s + (−0.399 + 0.692i)6-s + (−0.529 − 0.916i)7-s + (−0.481 − 0.604i)8-s + (−0.0388 − 0.518i)9-s + (−0.415 − 1.05i)10-s + (−0.162 − 0.0783i)11-s + (0.218 + 0.0674i)12-s + (−0.159 + 0.406i)13-s + (−0.894 + 0.830i)14-s + (−0.564 − 0.384i)15-s + (−0.761 + 0.954i)16-s + (0.872 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.438232 + 0.979343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438232 + 0.979343i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.44e5 + 3.91e5i)T \) |
| good | 2 | \( 1 + (2.90 + 12.7i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (23.7 + 22.0i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-272. + 41.0i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (480. + 831. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (717. + 345. i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (1.26e3 - 3.22e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-1.76e4 - 2.66e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (947. - 1.26e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (-3.42e4 + 2.33e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (6.87e4 - 6.37e4i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (1.87e5 + 5.78e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (-4.64e4 + 8.04e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-6.34e4 - 2.78e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (1.44e5 - 6.95e4i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (2.51e5 + 6.39e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (-9.04e5 + 1.13e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (-2.37e6 + 7.33e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-2.54e5 + 3.40e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (3.85e6 + 2.63e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (1.38e6 - 3.53e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (2.09e6 + 3.63e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.34e5 + 3.10e5i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (-6.13e5 - 5.69e5i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-6.97e6 - 3.35e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−13.18917874150610273817486906905, −12.48961144385431060465626749296, −11.24224259440405866691490519596, −10.13459431759566299910581604632, −9.308421769179881594807231032817, −7.03084115378481467879409832837, −5.86384865220782295780186446011, −3.53329680133130396757894398911, −1.72228226939151852260656442408, −0.50315817234462725311774793184,
2.54218414637781507182233137921, 5.36677209517968287744952089435, 5.91926346023858421016132929997, 7.46364676541538677955815275919, 8.997949786449467475415076367955, 10.08923014404523540742192669931, 11.52773249731992963158183446399, 13.01661489018866025342006792685, 14.38987046203239549135851459731, 15.47078744111725661075713500890
