| L(s) = 1 | + (−4.20 − 5.79i)2-s + (1.60 + 4.94i)3-s + (−10.8 + 33.5i)4-s + (33.2 + 24.1i)5-s + (21.8 − 30.0i)6-s + (−32.2 − 10.4i)7-s + (131. − 42.6i)8-s + (−21.8 + 15.8i)9-s − 294. i·10-s + (107. + 55.5i)11-s − 183.·12-s + (137. + 189. i)13-s + (74.9 + 230. i)14-s + (−66.0 + 203. i)15-s + (−342. − 248. i)16-s + (159. − 219. i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.44i)2-s + (0.178 + 0.549i)3-s + (−0.680 + 2.09i)4-s + (1.33 + 0.966i)5-s + (0.607 − 0.836i)6-s + (−0.657 − 0.213i)7-s + (2.04 − 0.665i)8-s + (−0.269 + 0.195i)9-s − 2.94i·10-s + (0.888 + 0.458i)11-s − 1.27·12-s + (0.814 + 1.12i)13-s + (0.382 + 1.17i)14-s + (−0.293 + 0.902i)15-s + (−1.33 − 0.971i)16-s + (0.550 − 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0928i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.976414 - 0.0454320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.976414 - 0.0454320i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.60 - 4.94i)T \) |
| 11 | \( 1 + (-107. - 55.5i)T \) |
| good | 2 | \( 1 + (4.20 + 5.79i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (-33.2 - 24.1i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (32.2 + 10.4i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (-137. - 189. i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-159. + 219. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (349. - 113. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + 66.9T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-219. - 71.4i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (111. - 81.1i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (-68.1 + 209. i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (1.51e3 - 493. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 + 2.64e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-218. - 673. i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-3.60e3 + 2.61e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-1.52e3 + 4.69e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (1.31e3 - 1.80e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 - 4.33e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (5.02e3 + 3.65e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-2.53e3 - 822. i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (2.32e3 + 3.19e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-2.38e3 + 3.28e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 - 1.18e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.20e3 + 2.32e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−16.50981009880967346877199363120, −14.43277185500410178419720213828, −13.36674341934642812061517251168, −11.77296529984086648858824329907, −10.53747210119817819430348781831, −9.789705537799881123416352512819, −8.956847020159443204914668750605, −6.65016209342284965872645070842, −3.60155633621495569697847187851, −2.00033290216914416008484427764,
1.10514605943216836828306566584, 5.72957851667510036885158181767, 6.39903239345858191238336983245, 8.310685712221989184187070372379, 9.063036800205556935529619746002, 10.21606817686977535901451935212, 12.78711217335840064371057793839, 13.75149006300678082086345465913, 15.02474943887627123281032886071, 16.37920486318441161411053286026
