| L(s) = 1 | + (−0.529 + 1.31i)2-s + (0.109 + 0.0767i)3-s + (−1.43 − 1.38i)4-s + (−1.98 + 1.03i)5-s + (−0.158 + 0.103i)6-s + (−2.66 + 0.713i)7-s + (2.58 − 1.15i)8-s + (−1.01 − 2.80i)9-s + (−0.300 − 3.14i)10-s + (2.35 − 1.36i)11-s + (−0.0510 − 0.262i)12-s + (1.29 − 0.904i)13-s + (0.475 − 3.87i)14-s + (−0.296 − 0.0392i)15-s + (0.139 + 3.99i)16-s + (5.84 − 2.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.374 + 0.927i)2-s + (0.0632 + 0.0443i)3-s + (−0.719 − 0.694i)4-s + (−0.887 + 0.461i)5-s + (−0.0648 + 0.0420i)6-s + (−1.00 + 0.269i)7-s + (0.913 − 0.406i)8-s + (−0.339 − 0.934i)9-s + (−0.0950 − 0.995i)10-s + (0.711 − 0.410i)11-s + (−0.0147 − 0.0758i)12-s + (0.358 − 0.250i)13-s + (0.127 − 1.03i)14-s + (−0.0766 − 0.0101i)15-s + (0.0348 + 0.999i)16-s + (1.41 − 0.660i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.447360 - 0.216852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.447360 - 0.216852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.529 - 1.31i)T \) |
| 5 | \( 1 + (1.98 - 1.03i)T \) |
| 19 | \( 1 + (2.72 + 3.39i)T \) |
| good | 3 | \( 1 + (-0.109 - 0.0767i)T + (1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (2.66 - 0.713i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.35 + 1.36i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 0.904i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-5.84 + 2.72i)T + (10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (6.14 - 0.537i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (3.08 + 8.47i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.73 + 2.15i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.895 + 0.895i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.00693 - 0.0393i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.839 - 9.59i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (2.29 + 1.07i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.0337 - 0.386i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.750 + 0.273i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.81 - 3.19i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.72 - 1.26i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (8.20 + 9.77i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.00 + 7.14i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 6.30i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.88 + 14.4i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (9.92 + 1.75i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (15.2 - 7.12i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.24529492495989644901680901998, −9.926381463812917428259896097239, −9.356433577444547195332790637698, −8.348333813307861308148965583049, −7.47164872868392855325976159638, −6.39320664096359261382012659317, −5.89694190789332407690209890331, −4.14149971804840660346955829904, −3.23524099398633702676361212926, −0.38804918344680379460148710595,
1.63784876583665396944549909546, 3.43640682795578882835049661843, 4.04823220385186355113219485990, 5.47388355675891943842773394925, 7.06296112455310415386453337911, 8.054193520314305755739333957313, 8.727765431613654216682382056281, 9.825947552558226521022938123855, 10.54402788455477041455082413203, 11.48329347651939351135660242116
