| L(s) = 1 | + (−21.7 + 15.8i)3-s − 26.6·5-s + (−46.2 − 142. i)7-s + (148. − 456. i)9-s + (53.6 + 165. i)11-s + (684. − 497. i)13-s + (580. − 421. i)15-s + (−444. + 1.36e3i)17-s + (−2.42e3 − 1.76e3i)19-s + (3.25e3 + 2.36e3i)21-s + (−706. + 2.17e3i)23-s − 2.41e3·25-s + (1.96e3 + 6.05e3i)27-s + (3.73e3 + 2.71e3i)29-s + (3.83e3 + 3.72e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.39 + 1.01i)3-s − 0.477·5-s + (−0.356 − 1.09i)7-s + (0.610 − 1.87i)9-s + (0.133 + 0.411i)11-s + (1.12 − 0.815i)13-s + (0.666 − 0.483i)15-s + (−0.372 + 1.14i)17-s + (−1.54 − 1.12i)19-s + (1.61 + 1.17i)21-s + (−0.278 + 0.857i)23-s − 0.772·25-s + (0.519 + 1.59i)27-s + (0.825 + 0.599i)29-s + (0.717 + 0.696i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7432338066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7432338066\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-3.83e3 - 3.72e3i)T \) |
| good | 3 | \( 1 + (21.7 - 15.8i)T + (75.0 - 231. i)T^{2} \) |
| 5 | \( 1 + 26.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + (46.2 + 142. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-53.6 - 165. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-684. + 497. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (444. - 1.36e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (2.42e3 + 1.76e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (706. - 2.17e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.73e3 - 2.71e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 - 4.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.30e3 - 6.75e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + (1.49e3 + 1.08e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-1.28e4 + 9.34e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-5.99e3 + 1.84e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.92e4 - 2.12e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 - 3.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.53e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.41e4 - 4.34e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-7.53e3 - 2.31e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (1.41e4 - 4.37e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.77e4 - 4.19e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-1.02e4 - 3.15e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.32e4 - 1.02e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.51619710161085974579092655049, −11.26897941547795127192812192177, −10.68521679522809527147103406665, −9.979786460978136738774764184972, −8.468420664592267890133286107136, −6.85675908327938466747589963549, −5.93200459757309420415718347757, −4.46166716552089570822502980256, −3.77774233744932408320964788919, −0.76662790795683767282002834536,
0.51410214698436086732060210234, 2.17588966821251850375106679454, 4.37859050395258406928986921242, 6.10899149907827996240179998721, 6.25471447308745948068536532727, 7.82100046431924306609757418150, 8.952982695520726504738857478413, 10.61623459545913861127317648746, 11.62160369456390233707150902447, 12.06401687395242484486703024367
