L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.22 − 2.12i)5-s + (−1 − 2.44i)7-s − 0.999i·8-s + (−2.12 − 1.22i)10-s + (0.866 + 0.5i)11-s − 2.44i·13-s + (−2.09 − 1.62i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + (−1.5 + 0.866i)19-s − 2.44·20-s + 0.999·22-s + (1.07 − 0.621i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.377 − 0.925i)7-s − 0.353i·8-s + (−0.670 − 0.387i)10-s + (0.261 + 0.150i)11-s − 0.679i·13-s + (−0.558 − 0.433i)14-s + (−0.125 − 0.216i)16-s + (−0.210 + 0.363i)17-s + (−0.344 + 0.198i)19-s − 0.547·20-s + 0.213·22-s + (0.224 − 0.129i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.392003855\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392003855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| 11 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.866 - 1.5i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 0.621i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.24 + 4.18i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-1.37 - 2.37i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.30 + 3.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.24 - 3.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.24iT - 71T^{2} \) |
| 73 | \( 1 + (4.24 + 2.44i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + (2.95 + 5.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.63iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.189198560802795072576082562910, −8.427163411179564665947872363881, −7.48130012888716245665955463149, −6.73130162268969195453403922634, −5.64998398299287637559942394209, −4.79190309576994620401885710852, −3.99424986439637633818664415403, −3.29817865962143968661886284977, −1.71383840539576619857320470764, −0.44466646345711225123551437892,
2.13940576002330532491381338527, 3.11297558409585762684898284266, 3.88605658658334053229738337829, 4.99167157690964242507969933299, 5.92224174511206266815549287888, 6.76433265609683614912330239015, 7.19491548260671692482028393724, 8.339159387658044886129729695139, 9.012500840582607313796382934456, 9.954998787328492332885108651612