| L(s) = 1 | + (−1.64 − 0.550i)3-s + (−1.00 − 2.76i)5-s + (−0.201 + 0.0355i)7-s + (2.39 + 1.80i)9-s + (−2.91 − 1.05i)11-s + (1.23 + 1.03i)13-s + (0.130 + 5.09i)15-s + (−2.86 − 1.65i)17-s + (−5.98 + 3.45i)19-s + (0.350 + 0.0525i)21-s + (−1.21 + 6.89i)23-s + (−2.79 + 2.34i)25-s + (−2.93 − 4.28i)27-s + (−2.70 − 3.22i)29-s + (0.173 + 0.0305i)31-s + ⋯ |
| L(s) = 1 | + (−0.948 − 0.317i)3-s + (−0.449 − 1.23i)5-s + (−0.0760 + 0.0134i)7-s + (0.797 + 0.602i)9-s + (−0.877 − 0.319i)11-s + (0.342 + 0.287i)13-s + (0.0336 + 1.31i)15-s + (−0.694 − 0.401i)17-s + (−1.37 + 0.792i)19-s + (0.0764 + 0.0114i)21-s + (−0.253 + 1.43i)23-s + (−0.559 + 0.469i)25-s + (−0.564 − 0.825i)27-s + (−0.503 − 0.599i)29-s + (0.0311 + 0.00548i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0326179 + 0.165706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0326179 + 0.165706i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.64 + 0.550i)T \) |
| good | 5 | \( 1 + (1.00 + 2.76i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.201 - 0.0355i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (2.91 + 1.05i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 1.03i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.86 + 1.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.98 - 3.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.21 - 6.89i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.70 + 3.22i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.173 - 0.0305i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (5.46 - 9.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.982 - 1.17i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.57 + 9.83i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.76 + 9.98i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.56iT - 53T^{2} \) |
| 59 | \( 1 + (-4.69 + 1.71i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 6.69i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.60 + 3.10i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.61 - 4.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.68 + 8.11i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.13 - 7.31i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.47 + 7.10i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.64 + 1.52i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 + 0.577i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.79832243656747131818082200554, −9.861973911421910549171225316894, −8.641205471867678611725335060792, −7.981366408760120313981390932610, −6.83411492860211619711017403929, −5.72378448454070405314968298105, −4.94264172946306605198258808532, −3.92850053349949564907059479909, −1.78467246903575059683799426518, −0.11559075456755462889056706579,
2.48968103148045919284648784857, 3.87986295441084953497271600575, 4.88970112355368868003080179879, 6.22777089941730038999844152360, 6.78426929935124139900287287231, 7.81205391737695003822645766527, 9.041285448519502948790734024636, 10.35405198542527159889925037540, 10.79134205357422821666893594343, 11.24408685239286620058823105335
