| 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
−11.24408685239286620058823105335, −10.79134205357422821666893594343, −10.35405198542527159889925037540, −9.041285448519502948790734024636, −7.81205391737695003822645766527, −6.78426929935124139900287287231, −6.22777089941730038999844152360, −4.88970112355368868003080179879, −3.87986295441084953497271600575, −2.48968103148045919284648784857,
0.11559075456755462889056706579, 1.78467246903575059683799426518, 3.92850053349949564907059479909, 4.94264172946306605198258808532, 5.72378448454070405314968298105, 6.83411492860211619711017403929, 7.981366408760120313981390932610, 8.641205471867678611725335060792, 9.861973911421910549171225316894, 10.79832243656747131818082200554
