| 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
−15.09285939212921306159161718308, −13.58992912271115851229095788623, −12.58423908655240708588445609594, −11.87552493464761053449843235936, −10.05141295254237724985299164407, −8.640932597950524197951361791616, −7.900700021221012596095112922575, −5.70837824350145769561165268461, −3.35978524632558827079567746913, −2.17762340252416254399856007848,
0.092386844695323241415476546180, 3.37080099502630550904467994021, 4.50176182809455892932795150772, 6.93145342011780384147036770714, 7.75675991271323728697195892158, 9.561513336156057148489324399369, 10.12761734724311761946899019332, 12.11437583594139205495345829931, 13.87899558401761451441203847043, 14.37818696653444134429495498235
