| L(s) = 1 | + (−7.27 + 8.66i)2-s + (−74.4 − 31.9i)3-s + (−22.2 − 126. i)4-s + (−303. + 833. i)5-s + (818. − 412. i)6-s + (−11.0 + 62.7i)7-s + (1.25e3 + 724. i)8-s + (4.52e3 + 4.75e3i)9-s + (−5.01e3 − 8.68e3i)10-s + (9.13e3 + 2.50e4i)11-s + (−2.37e3 + 1.00e4i)12-s + (1.85e4 − 1.55e4i)13-s + (−463. − 552. i)14-s + (4.91e4 − 5.23e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (−6.81e4 + 3.93e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.918 − 0.394i)3-s + (−0.0868 − 0.492i)4-s + (−0.485 + 1.33i)5-s + (0.631 − 0.318i)6-s + (−0.00461 + 0.0261i)7-s + (0.306 + 0.176i)8-s + (0.689 + 0.724i)9-s + (−0.501 − 0.868i)10-s + (0.623 + 1.71i)11-s + (−0.114 + 0.486i)12-s + (0.650 − 0.545i)13-s + (−0.0120 − 0.0143i)14-s + (0.971 − 1.03i)15-s + (−0.234 + 0.0855i)16-s + (−0.816 + 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.134806 - 0.310011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134806 - 0.310011i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.27 - 8.66i)T \) |
| 3 | \( 1 + (74.4 + 31.9i)T \) |
| good | 5 | \( 1 + (303. - 833. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (11.0 - 62.7i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-9.13e3 - 2.50e4i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-1.85e4 + 1.55e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (6.81e4 - 3.93e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.69e4 - 1.33e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (5.40e4 - 9.53e3i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (1.55e4 - 1.84e4i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (2.84e5 + 1.61e6i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-5.91e5 - 1.02e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-6.08e4 - 7.25e4i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (2.97e6 - 1.08e6i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (5.66e6 + 9.99e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 7.83e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (6.93e6 - 1.90e7i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-1.42e5 + 8.06e5i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (9.66e6 - 8.11e6i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-3.63e7 + 2.09e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (8.83e6 - 1.53e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (3.43e7 + 2.88e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-4.76e7 + 5.67e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-3.61e7 - 2.08e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (5.20e7 - 1.89e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−14.82436536019374369458988981074, −13.16372772139312463138006678858, −11.85583991857964082672043466852, −10.80488921783851465393497293736, −9.918648637296999513417484302853, −7.920258952094509600995421562661, −6.92642098978640571805927976962, −6.10638679797207151165062898788, −4.22101104096222851679562106197, −1.85076288852598622546265636855,
0.18405185792382956737479655291, 1.13193427113882348198192896499, 3.75936376892411747961505169905, 4.94305027211996878129861219755, 6.53869086288501055683229102861, 8.585476618130151297657613328505, 9.149174509344524948012225118259, 10.92043719664393673942089240707, 11.52315687226221980210961285274, 12.58244218387580028665557484845
