| L(s) = 1 | + (2.65 + 0.278i)2-s + (−0.743 + 0.669i)3-s + (5.01 + 1.06i)4-s + (−0.0958 + 0.215i)5-s + (−2.15 + 1.56i)6-s + (−2.64 + 0.107i)7-s + (7.92 + 2.57i)8-s + (0.104 − 0.994i)9-s + (−0.314 + 0.544i)10-s + (−1.24 − 3.07i)11-s + (−4.43 + 2.56i)12-s + (−3.01 − 2.19i)13-s + (−7.04 − 0.451i)14-s + (−0.0728 − 0.224i)15-s + (10.9 + 4.87i)16-s + (0.401 + 3.82i)17-s + ⋯ |
| L(s) = 1 | + (1.87 + 0.197i)2-s + (−0.429 + 0.386i)3-s + (2.50 + 0.532i)4-s + (−0.0428 + 0.0962i)5-s + (−0.881 + 0.640i)6-s + (−0.999 + 0.0406i)7-s + (2.80 + 0.910i)8-s + (0.0348 − 0.331i)9-s + (−0.0994 + 0.172i)10-s + (−0.376 − 0.926i)11-s + (−1.28 + 0.739i)12-s + (−0.836 − 0.607i)13-s + (−1.88 − 0.120i)14-s + (−0.0187 − 0.0578i)15-s + (2.73 + 1.21i)16-s + (0.0974 + 0.927i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65887 + 0.806158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65887 + 0.806158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.743 - 0.669i)T \) |
| 7 | \( 1 + (2.64 - 0.107i)T \) |
| 11 | \( 1 + (1.24 + 3.07i)T \) |
| good | 2 | \( 1 + (-2.65 - 0.278i)T + (1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (0.0958 - 0.215i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (3.01 + 2.19i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.401 - 3.82i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-4.68 + 0.995i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (2.07 + 3.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.14 - 1.02i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.04 + 6.83i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (2.80 - 3.11i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.45 - 4.48i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (0.775 + 3.64i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (5.81 - 2.59i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.532 + 2.50i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.48 - 2.88i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-5.39 + 9.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.72 - 1.97i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 2.55i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-10.9 - 1.14i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (12.9 - 9.39i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (7.15 - 4.13i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.40 + 1.93i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.66281218244675060646035969877, −11.55611275510117461477296961726, −10.79472277574939232213501050935, −9.725346695860819707724839387902, −7.895383331337047177974931591116, −6.71814310913799258170427800791, −5.85187582783322204243546443376, −5.07334561861514051164212815461, −3.70420623945121486626662350265, −2.87075917432225427918987379517,
2.20696746460176022711463959517, 3.51046220974468785354590673458, 4.85277681731388446391989054646, 5.60532737929371074008635976610, 6.94029031999726846200991252224, 7.27761977467468257492307455560, 9.582927562779232308280728318440, 10.52442130072543797238025412452, 11.77762785674314365083008420904, 12.25127358875106251648886401348
