| L(s) = 1 | + (3.46 − 1.12i)2-s + (20.0 − 18.1i)3-s + (−41.0 + 29.8i)4-s + (−207. − 67.4i)5-s + (48.9 − 85.3i)6-s + (−309. + 224. i)7-s + (−245. + 338. i)8-s + (71.5 − 725. i)9-s − 794.·10-s + (1.31e3 − 202. i)11-s + (−280. + 1.34e3i)12-s + (−49.3 − 152. i)13-s + (−819. + 1.12e3i)14-s + (−5.37e3 + 2.41e3i)15-s + (532. − 1.63e3i)16-s + (−2.81e3 − 914. i)17-s + ⋯ |
| L(s) = 1 | + (0.433 − 0.140i)2-s + (0.740 − 0.671i)3-s + (−0.641 + 0.465i)4-s + (−1.65 − 0.539i)5-s + (0.226 − 0.395i)6-s + (−0.901 + 0.655i)7-s + (−0.479 + 0.660i)8-s + (0.0981 − 0.995i)9-s − 0.794·10-s + (0.988 − 0.152i)11-s + (−0.162 + 0.775i)12-s + (−0.0224 − 0.0691i)13-s + (−0.298 + 0.410i)14-s + (−1.59 + 0.714i)15-s + (0.129 − 0.399i)16-s + (−0.572 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0988i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.995 - 0.0988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0170055 + 0.343394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0170055 + 0.343394i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-20.0 + 18.1i)T \) |
| 11 | \( 1 + (-1.31e3 + 202. i)T \) |
| good | 2 | \( 1 + (-3.46 + 1.12i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (207. + 67.4i)T + (1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (309. - 224. i)T + (3.63e4 - 1.11e5i)T^{2} \) |
| 13 | \( 1 + (49.3 + 152. i)T + (-3.90e6 + 2.83e6i)T^{2} \) |
| 17 | \( 1 + (2.81e3 + 914. i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (8.90e3 + 6.47e3i)T + (1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + 3.86e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + (-650. - 895. i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-9.41e3 - 2.89e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (3.22e4 - 2.34e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (3.52e3 - 4.85e3i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 8.88e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-6.28e4 + 8.65e4i)T + (-3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (1.55e4 - 5.04e3i)T + (1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (3.17e4 + 4.37e4i)T + (-1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 + (1.15e5 - 3.54e5i)T + (-4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 - 3.72e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-6.48e4 - 2.10e4i)T + (1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-4.12e4 + 2.99e4i)T + (4.67e10 - 1.43e11i)T^{2} \) |
| 79 | \( 1 + (2.24e5 + 6.90e5i)T + (-1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (6.96e5 + 2.26e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 + 1.34e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (1.96e5 + 6.04e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−14.77706739072543157815601273799, −13.30518337415084093837728922992, −12.43662730846902074053507835903, −11.74984636262835494855879556923, −8.949984895972362902227495786900, −8.483330093155744050332599733552, −6.83859280753726522406656491063, −4.32050437501706729089830922386, −3.13842880199495404679456758359, −0.14446247493492639700394819991,
3.67925199761086328242904195597, 4.19862468251685715864348633594, 6.71359909497767161223056650422, 8.311062966096699885058412735525, 9.693751708636234290163018335456, 10.91149578675615028043643756474, 12.58877830129581730799524364355, 13.95625268810901576269095410141, 14.92618854321373944592673350175, 15.59200856713570660573509387777
