| L(s) = 1 | + (−1.30 + 0.552i)2-s + (1.55 + 2.02i)3-s + (1.38 − 1.43i)4-s + (0.189 + 0.145i)5-s + (−3.13 − 1.77i)6-s + (−1.52 + 2.16i)7-s + (−1.01 + 2.64i)8-s + (−0.904 + 3.37i)9-s + (−0.326 − 0.0844i)10-s + (3.61 + 0.475i)11-s + (5.06 + 0.577i)12-s + (−1.86 − 0.773i)13-s + (0.789 − 3.65i)14-s + 0.607i·15-s + (−0.138 − 3.99i)16-s + (0.603 + 0.348i)17-s + ⋯ |
| L(s) = 1 | + (−0.920 + 0.390i)2-s + (0.895 + 1.16i)3-s + (0.694 − 0.719i)4-s + (0.0846 + 0.0649i)5-s + (−1.28 − 0.724i)6-s + (−0.576 + 0.817i)7-s + (−0.358 + 0.933i)8-s + (−0.301 + 1.12i)9-s + (−0.103 − 0.0267i)10-s + (1.08 + 0.143i)11-s + (1.46 + 0.166i)12-s + (−0.517 − 0.214i)13-s + (0.211 − 0.977i)14-s + 0.156i·15-s + (−0.0347 − 0.999i)16-s + (0.146 + 0.0844i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.597429 + 0.851940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.597429 + 0.851940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.552i)T \) |
| 7 | \( 1 + (1.52 - 2.16i)T \) |
| good | 3 | \( 1 + (-1.55 - 2.02i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-0.189 - 0.145i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (-3.61 - 0.475i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (1.86 + 0.773i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-0.603 - 0.348i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0186 + 0.141i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.87 - 6.98i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.11 - 2.68i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-3.54 + 6.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.48 + 3.43i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-6.01 - 6.01i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.31 + 3.18i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-9.59 + 5.54i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.83 + 0.768i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-1.19 + 9.06i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-5.97 + 0.786i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-8.96 - 11.6i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-9.68 + 9.68i)T - 71iT^{2} \) |
| 73 | \( 1 + (-10.3 + 2.77i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.926 - 0.534i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.15 + 0.893i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (10.9 + 2.93i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−12.31395124354222111115835513328, −11.32666701337912711007529804972, −10.01083853502099419828853712418, −9.585872978822499144234662409397, −8.892072498890144258079027301206, −7.917697351914249896565558344640, −6.55744414954202052116687345018, −5.38135368455183692762711670664, −3.76563559237738170784664021537, −2.37730844975651392549533189062,
1.16168673725687002421636583116, 2.59461535930904283592650167788, 3.88122569532302275897966892094, 6.50487357815323933150669326793, 7.07886457752947404513253132644, 8.033781562161511235748644613092, 8.966505486618619586840135390147, 9.790871600109027441498359277853, 10.91370236061094915612024902625, 12.18279697897483141150434938752
