| L(s) = 1 | + (−3.39 + 1.96i)5-s + (−6.39 + 11.0i)7-s + (−14.2 − 8.25i)11-s + (−1.39 − 2.42i)13-s − 2.54i·17-s − 21.5·19-s + (−2.60 + 1.50i)23-s + (−4.79 + 8.31i)25-s + (−13.1 − 7.61i)29-s + (13.6 + 23.5i)31-s − 50.2i·35-s + 10.4·37-s + (34.5 − 19.9i)41-s + (17.0 − 29.6i)43-s + (−58.1 − 33.6i)47-s + ⋯ |
| L(s) = 1 | + (−0.679 + 0.392i)5-s + (−0.914 + 1.58i)7-s + (−1.29 − 0.750i)11-s + (−0.107 − 0.186i)13-s − 0.149i·17-s − 1.13·19-s + (−0.113 + 0.0652i)23-s + (−0.191 + 0.332i)25-s + (−0.455 − 0.262i)29-s + (0.438 + 0.759i)31-s − 1.43i·35-s + 0.281·37-s + (0.841 − 0.485i)41-s + (0.397 − 0.688i)43-s + (−1.23 − 0.714i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4817063744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4817063744\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.39 - 1.96i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (6.39 - 11.0i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (14.2 + 8.25i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.39 + 2.42i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 2.54iT - 289T^{2} \) |
| 19 | \( 1 + 21.5T + 361T^{2} \) |
| 23 | \( 1 + (2.60 - 1.50i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.1 + 7.61i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-13.6 - 23.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 10.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-17.0 + 29.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (58.1 + 33.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-5.29 + 3.05i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.3 + 35.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-54.4 - 94.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 52.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-12.7 + 22.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-52.0 - 30.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 7.62iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-50.7 + 87.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−8.910880372004163638036437171633, −8.343371887337811970464914160458, −7.56042990914203305332323579852, −6.54965055892769027593061059423, −5.79689869101160703263901050923, −5.16171857271938035107179669192, −3.81717122987854392817187423575, −2.92046249838482300599648958405, −2.31103504013228149963099954614, −0.20772264429018792308871420732,
0.62497047715582913213327895865, 2.22047351622039785182921293633, 3.44749989445467143541125607870, 4.27204452873666910539818769149, 4.82388618778976018918412192627, 6.20390789602903905188883976657, 6.88649629236398920857790156436, 7.83327936376628648674196482148, 8.050251650864724049198411308963, 9.428794280761766845727361731072
