| 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
−12.17235912622950883342152694331, −11.09045998598942588306435670686, −10.43903022636811844938318366878, −9.918911556972698702996600690302, −9.058154687265618097114755520408, −7.63792340601803975130725361389, −6.39073262466009698925420502179, −4.93828999612309752457992969694, −2.88341965333544808844160185803, −1.35023401729510990645638386741,
0.996110629027638306056963217360, 1.82866480454983765098024135452, 5.50401134338940802620413030391, 5.87914808295780209555466077259, 7.37937423332186570636785491548, 8.367228606175155070593138224054, 9.090046670332836571224494235618, 10.11488603075986649642234528005, 11.30876958630046352818114302686, 12.31121592098450437053756019984
