| L(s) = 1 | + (−4.20 + 1.64i)2-s + (−2.59 − 4.50i)3-s + (9.07 − 8.41i)4-s + (15.6 − 10.6i)5-s + (18.3 + 14.6i)6-s + (16.0 + 9.25i)7-s + (−8.57 + 17.7i)8-s + (−13.5 + 23.3i)9-s + (−48.0 + 70.4i)10-s + (4.73 − 31.3i)11-s + (−61.4 − 19.0i)12-s + (13.7 + 10.9i)13-s + (−82.6 − 12.4i)14-s + (−88.3 − 42.6i)15-s + (−0.734 + 9.80i)16-s + (50.1 − 15.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.48 + 0.583i)2-s + (−0.499 − 0.866i)3-s + (1.13 − 1.05i)4-s + (1.39 − 0.951i)5-s + (1.24 + 0.996i)6-s + (0.866 + 0.499i)7-s + (−0.378 + 0.786i)8-s + (−0.501 + 0.864i)9-s + (−1.51 + 2.22i)10-s + (0.129 − 0.860i)11-s + (−1.47 − 0.457i)12-s + (0.294 + 0.234i)13-s + (−1.57 − 0.237i)14-s + (−1.52 − 0.734i)15-s + (−0.0114 + 0.153i)16-s + (0.715 − 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.889975 - 0.377205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.889975 - 0.377205i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.59 + 4.50i)T \) |
| 7 | \( 1 + (-16.0 - 9.25i)T \) |
| good | 2 | \( 1 + (4.20 - 1.64i)T + (5.86 - 5.44i)T^{2} \) |
| 5 | \( 1 + (-15.6 + 10.6i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-4.73 + 31.3i)T + (-1.27e3 - 392. i)T^{2} \) |
| 13 | \( 1 + (-13.7 - 10.9i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-50.1 + 15.4i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-87.2 - 50.3i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (35.9 - 116. i)T + (-1.00e4 - 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-5.99 - 1.36i)T + (2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-95.2 + 54.9i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (296. + 275. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-44.3 - 21.3i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-452. + 217. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (120. + 307. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (289. + 312. i)T + (-1.11e4 + 1.48e5i)T^{2} \) |
| 59 | \( 1 + (76.0 + 51.8i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (292. - 315. i)T + (-1.69e4 - 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-389. - 675. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (431. - 98.5i)T + (3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (784. + 307. i)T + (2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (128. - 222. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-382. - 479. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-244. + 36.9i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 861. iT - 9.12e5T^{2} \) |
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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
−12.31121592098450437053756019984, −11.30876958630046352818114302686, −10.11488603075986649642234528005, −9.090046670332836571224494235618, −8.367228606175155070593138224054, −7.37937423332186570636785491548, −5.87914808295780209555466077259, −5.50401134338940802620413030391, −1.82866480454983765098024135452, −0.996110629027638306056963217360,
1.35023401729510990645638386741, 2.88341965333544808844160185803, 4.93828999612309752457992969694, 6.39073262466009698925420502179, 7.63792340601803975130725361389, 9.058154687265618097114755520408, 9.918911556972698702996600690302, 10.43903022636811844938318366878, 11.09045998598942588306435670686, 12.17235912622950883342152694331
