| L(s) = 1 | + (−1.38 + 7.87i)2-s + (37.7 + 31.6i)3-s + (−60.1 − 21.8i)4-s + (−192. + 70.0i)5-s + (−302. + 253. i)6-s + (−814. − 1.41e3i)7-s + (256 − 443. i)8-s + (41.9 + 237. i)9-s + (−284. − 1.61e3i)10-s + (−1.68e3 + 2.92e3i)11-s + (−1.57e3 − 2.73e3i)12-s + (−6.31e3 + 5.29e3i)13-s + (1.22e4 − 4.45e3i)14-s + (−9.48e3 − 3.45e3i)15-s + (3.13e3 + 2.63e3i)16-s + (6.35e3 − 3.60e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.807 + 0.677i)3-s + (−0.469 − 0.171i)4-s + (−0.688 + 0.250i)5-s + (−0.570 + 0.478i)6-s + (−0.897 − 1.55i)7-s + (0.176 − 0.306i)8-s + (0.0191 + 0.108i)9-s + (−0.0899 − 0.510i)10-s + (−0.382 + 0.662i)11-s + (−0.263 − 0.456i)12-s + (−0.796 + 0.668i)13-s + (1.19 − 0.434i)14-s + (−0.725 − 0.264i)15-s + (0.191 + 0.160i)16-s + (0.313 − 1.77i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.148222 - 0.181865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148222 - 0.181865i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 7.87i)T \) |
| 19 | \( 1 + (-2.26e4 - 1.95e4i)T \) |
| good | 3 | \( 1 + (-37.7 - 31.6i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (192. - 70.0i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (814. + 1.41e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.68e3 - 2.92e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.31e3 - 5.29e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-6.35e3 + 3.60e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (1.02e5 + 3.71e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-6.90e3 - 3.91e4i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-5.40e4 - 9.35e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.26e3 + 4.42e3i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (9.58e4 - 3.48e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.23e5 - 7.03e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.10e6 + 4.02e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-1.90e5 + 1.08e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.65e6 - 6.03e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (3.85e5 + 2.18e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (4.15e6 - 1.51e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (3.28e5 + 2.75e5i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-6.28e5 - 5.27e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-5.32e5 - 9.21e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-5.64e6 + 4.73e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-6.34e5 + 3.60e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.37818696653444134429495498235, −13.87899558401761451441203847043, −12.11437583594139205495345829931, −10.12761734724311761946899019332, −9.561513336156057148489324399369, −7.75675991271323728697195892158, −6.93145342011780384147036770714, −4.50176182809455892932795150772, −3.37080099502630550904467994021, −0.092386844695323241415476546180,
2.17762340252416254399856007848, 3.35978524632558827079567746913, 5.70837824350145769561165268461, 7.900700021221012596095112922575, 8.640932597950524197951361791616, 10.05141295254237724985299164407, 11.87552493464761053449843235936, 12.58423908655240708588445609594, 13.58992912271115851229095788623, 15.09285939212921306159161718308
