| L(s) = 1 | + (−7.80 + 5.67i)2-s + (14.8 − 4.69i)3-s + (18.8 − 58.0i)4-s + (−46.3 + 63.7i)5-s + (−89.4 + 120. i)6-s + (−135. − 43.9i)7-s + (86.6 + 266. i)8-s + (198. − 139. i)9-s − 759. i·10-s + (−281. − 286. i)11-s + (7.91 − 951. i)12-s + (−87.0 − 119. i)13-s + (1.30e3 − 424. i)14-s + (−389. + 1.16e3i)15-s + (−607. − 441. i)16-s + (−1.75e3 − 1.27e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 + 1.00i)2-s + (0.953 − 0.301i)3-s + (0.589 − 1.81i)4-s + (−0.828 + 1.14i)5-s + (−1.01 + 1.37i)6-s + (−1.04 − 0.339i)7-s + (0.478 + 1.47i)8-s + (0.818 − 0.574i)9-s − 2.40i·10-s + (−0.700 − 0.713i)11-s + (0.0158 − 1.90i)12-s + (−0.142 − 0.196i)13-s + (1.78 − 0.578i)14-s + (−0.446 + 1.33i)15-s + (−0.592 − 0.430i)16-s + (−1.47 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0365244 - 0.0562346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0365244 - 0.0562346i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.8 + 4.69i)T \) |
| 11 | \( 1 + (281. + 286. i)T \) |
| good | 2 | \( 1 + (7.80 - 5.67i)T + (9.88 - 30.4i)T^{2} \) |
| 5 | \( 1 + (46.3 - 63.7i)T + (-965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (135. + 43.9i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (87.0 + 119. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.75e3 + 1.27e3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (95.9 - 31.1i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 368. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (1.42e3 - 4.40e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.09e3 - 2.24e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.57e3 - 7.91e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-4.26e3 - 1.31e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 488. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.06e4 + 3.46e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.02e4 + 2.78e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.49e4 - 4.84e3i)T + (5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.14e3 - 4.32e3i)T + (-2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 1.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.07e4 - 2.86e4i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.56e4 + 1.48e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.69e4 - 3.71e4i)T + (-9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.25e4 + 1.63e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 1.01e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.47e4 - 6.88e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.62227044885130298566823819469, −14.60014590237463204978255821386, −13.23468277811165449040789664458, −10.98834389273251421015268376220, −9.794056745186289125794416634937, −8.549200692397329655371364772287, −7.32504291874510769859445240640, −6.67576454120965406992230926798, −3.10331696927195042270891807322, −0.05304241455541078412430250925,
2.23691262007588698765854534568, 4.06040517140193875078893265545, 7.62040202047815013742114736771, 8.742644919694187368091257467725, 9.427092367277610751929775310596, 10.71079035829666298032571279954, 12.38884722248479561480451868369, 13.01560856844820556677615902088, 15.40468858788202948008574189030, 16.12722505547720178473259866284
