| L(s) = 1 | + (0.590 + 0.655i)2-s + (−1.65 − 0.504i)3-s + (0.127 − 1.21i)4-s + (−0.627 + 2.95i)5-s + (−0.647 − 1.38i)6-s + (2.53 − 0.745i)7-s + (2.29 − 1.67i)8-s + (2.49 + 1.67i)9-s + (−2.30 + 1.33i)10-s + (3.06 + 1.26i)11-s + (−0.824 + 1.94i)12-s + (1.56 + 0.506i)13-s + (1.98 + 1.22i)14-s + (2.52 − 4.57i)15-s + (0.0657 + 0.0139i)16-s + (0.0265 − 0.0294i)17-s + ⋯ |
| L(s) = 1 | + (0.417 + 0.463i)2-s + (−0.956 − 0.291i)3-s + (0.0638 − 0.607i)4-s + (−0.280 + 1.31i)5-s + (−0.264 − 0.565i)6-s + (0.959 − 0.281i)7-s + (0.813 − 0.590i)8-s + (0.830 + 0.557i)9-s + (−0.729 + 0.420i)10-s + (0.923 + 0.382i)11-s + (−0.238 + 0.562i)12-s + (0.432 + 0.140i)13-s + (0.531 + 0.327i)14-s + (0.652 − 1.18i)15-s + (0.0164 + 0.00349i)16-s + (0.00643 − 0.00715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26657 + 0.307534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26657 + 0.307534i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.504i)T \) |
| 7 | \( 1 + (-2.53 + 0.745i)T \) |
| 11 | \( 1 + (-3.06 - 1.26i)T \) |
| good | 2 | \( 1 + (-0.590 - 0.655i)T + (-0.209 + 1.98i)T^{2} \) |
| 5 | \( 1 + (0.627 - 2.95i)T + (-4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 0.506i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.0265 + 0.0294i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.72 - 0.181i)T + (18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-3.55 - 2.05i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.87 + 4.99i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.15 + 1.09i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (8.45 - 3.76i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (4.13 - 3.00i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.302iT - 43T^{2} \) |
| 47 | \( 1 + (0.405 - 0.0426i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.49 - 7.02i)T + (-48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (9.86 + 1.03i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-2.87 + 13.5i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (4.09 + 7.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.15 + 2.00i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.63 - 0.592i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.10 + 0.994i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-4.09 - 12.5i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (13.1 + 7.57i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.75 - 5.40i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−12.03784920315153041044844544159, −11.13307211626004581194327406128, −10.76561848316545971241407802157, −9.671112047399391233299859676238, −7.80475338950903450415434750849, −6.88162283472865869926723833940, −6.34312174064537194331198637164, −5.10580786913588634494094728101, −4.01005426271954108056151751995, −1.62907478562620794104799763087,
1.43200812774333857054927229879, 3.78935015541928094164191030441, 4.68143885902363418879948947342, 5.48471996427544837361253772460, 7.05677409712046546333335721505, 8.436183546033666638358792770663, 8.982237798974964160839893971891, 10.67616861251572747490597974468, 11.41339386695676668155579862597, 12.11665436496000501806608824926
