| L(s) = 1 | + (−1.23 − 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (11.2 − 54.7i)5-s + (−29.1 − 21.1i)6-s + 29.3·7-s + (51.7 + 37.6i)8-s + (25.0 − 77.0i)9-s + (−222. + 24.7i)10-s + (−77.0 − 237. i)11-s + (−44.4 + 136. i)12-s + (138. − 425. i)13-s + (−36.2 − 111. i)14-s + (−207. − 458. i)15-s + (79.1 − 243. i)16-s + (558. + 405. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.201 − 0.979i)5-s + (−0.330 − 0.239i)6-s + 0.226·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.702 + 0.0782i)10-s + (−0.191 − 0.590i)11-s + (−0.0892 + 0.274i)12-s + (0.226 − 0.698i)13-s + (−0.0493 − 0.152i)14-s + (−0.238 − 0.525i)15-s + (0.0772 − 0.237i)16-s + (0.468 + 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0294i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.539071335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539071335\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 + (-7.28 + 5.29i)T \) |
| 5 | \( 1 + (-11.2 + 54.7i)T \) |
| good | 7 | \( 1 - 29.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (77.0 + 237. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-138. + 425. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-558. - 405. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.00e3 + 730. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-22.6 - 69.6i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.57e3 + 1.14e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.17e3 + 2.30e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-17.6 + 54.3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.41e3 - 4.35e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 6.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.98e3 - 3.62e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.61e4 - 1.90e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-360. + 1.10e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-3.57e3 - 1.10e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-2.35e3 - 1.70e3i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.61e4 + 1.17e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.26e4 + 3.88e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-7.09e4 + 5.15e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.49e4 + 3.99e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-1.62e4 - 5.00e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.14e5 + 8.33e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.67650590773343802812153757447, −10.56653588733567653940678825327, −9.444589243523150081968471947495, −8.506472330000567167934037927575, −7.82361641383526816506164812157, −6.01006947364155071542481258468, −4.68065428226817764963609728747, −3.24487246803560273939480949102, −1.75510491229505510561384527122, −0.52016712724539298985494332529,
1.95106298669670031037745920512, 3.55338036105331639731986903051, 4.95249227352782310675748952999, 6.38060612793421747400181500418, 7.32019649038383144027748402609, 8.372033772874908886118586956756, 9.555467232242641702779084999033, 10.32784341306256388708005818669, 11.38631579446610847065898935345, 12.81858257420574041235230984277
