| L(s) = 1 | + (−7.78 + 5.65i)2-s + (−2.78 − 8.55i)3-s + (18.7 − 57.6i)4-s + (48.1 + 34.9i)5-s + (70.0 + 50.8i)6-s + (9.43 − 29.0i)7-s + (84.9 + 261. i)8-s + (−65.5 + 47.6i)9-s − 572.·10-s + (−70.0 + 395. i)11-s − 545.·12-s + (−460. + 334. i)13-s + (90.7 + 279. i)14-s + (165. − 509. i)15-s + (−572. − 415. i)16-s + (1.61e3 + 1.17e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.999i)2-s + (−0.178 − 0.549i)3-s + (0.584 − 1.80i)4-s + (0.861 + 0.625i)5-s + (0.794 + 0.577i)6-s + (0.0727 − 0.224i)7-s + (0.469 + 1.44i)8-s + (−0.269 + 0.195i)9-s − 1.81·10-s + (−0.174 + 0.984i)11-s − 1.09·12-s + (−0.756 + 0.549i)13-s + (0.123 + 0.381i)14-s + (0.189 − 0.584i)15-s + (−0.558 − 0.405i)16-s + (1.35 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.387610 + 0.603578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.387610 + 0.603578i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.78 + 8.55i)T \) |
| 11 | \( 1 + (70.0 - 395. i)T \) |
| good | 2 | \( 1 + (7.78 - 5.65i)T + (9.88 - 30.4i)T^{2} \) |
| 5 | \( 1 + (-48.1 - 34.9i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-9.43 + 29.0i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (460. - 334. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.61e3 - 1.17e3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-497. - 1.53e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 1.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (2.34e3 - 7.22e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-5.87e3 + 4.26e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (406. - 1.25e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.80e3 - 5.55e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 6.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (9.34e3 + 2.87e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (5.78e3 - 4.20e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (6.85e3 - 2.10e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.02e4 + 1.46e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.22e4 - 3.79e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.36e4 + 4.19e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.41e4 + 1.02e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (9.61e4 + 6.98e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 6.61e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.03e5 - 7.49e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−16.59746603058459849118824895839, −14.94705387486580056149984168018, −14.16909859265863090246233300121, −12.33963370808598988008440248661, −10.37705940499647739946661226758, −9.711799912387417569534374284947, −7.967729422045492351966213060583, −6.96420618358978185147401467455, −5.78640071834111103046752031554, −1.68811711634890474199752367383,
0.72211166929673124858496498961, 2.79252045874251224867094681138, 5.42881130654092828413559236530, 7.966917142004799275151121745855, 9.311245053756885801323357267796, 9.941696231560411302809827150449, 11.25208416679923897863839858885, 12.35313544539954706235701911110, 13.89441701280487809931063854416, 15.85982314821760886151782495043
