| L(s) = 1 | + (2.25 − 1.00i)2-s + (1.58 − 0.336i)3-s + (2.73 − 3.03i)4-s + (0.362 + 3.44i)5-s + (3.22 − 2.34i)6-s + (1.58 − 4.89i)8-s + (−0.348 + 0.155i)9-s + (4.27 + 7.40i)10-s + (−2.82 − 1.74i)11-s + (3.30 − 5.72i)12-s + (0.528 + 0.384i)13-s + (1.73 + 5.33i)15-s + (−0.472 − 4.49i)16-s + (−1.03 − 0.462i)17-s + (−0.630 + 0.700i)18-s + (−4.06 − 4.51i)19-s + ⋯ |
| L(s) = 1 | + (1.59 − 0.709i)2-s + (0.913 − 0.194i)3-s + (1.36 − 1.51i)4-s + (0.162 + 1.54i)5-s + (1.31 − 0.957i)6-s + (0.561 − 1.72i)8-s + (−0.116 + 0.0517i)9-s + (1.35 + 2.34i)10-s + (−0.850 − 0.525i)11-s + (0.953 − 1.65i)12-s + (0.146 + 0.106i)13-s + (0.447 + 1.37i)15-s + (−0.118 − 1.12i)16-s + (−0.251 − 0.112i)17-s + (−0.148 + 0.165i)18-s + (−0.932 − 1.03i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.01682 - 1.36710i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.01682 - 1.36710i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (2.82 + 1.74i)T \) |
| good | 2 | \( 1 + (-2.25 + 1.00i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-1.58 + 0.336i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.362 - 3.44i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-0.528 - 0.384i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.03 + 0.462i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (4.06 + 4.51i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.33 + 5.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.41 + 4.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.292 - 2.78i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.430 - 0.0914i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (1.82 - 5.61i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (0.404 + 0.449i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.02 - 9.76i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-1.13 + 1.25i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.716 + 6.82i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 5.35i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.38 + 3.18i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.48 + 4.98i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-2.42 + 1.07i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (5.41 - 3.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.349 + 0.604i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 - 8.73i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.89272494690098014183172504437, −10.43372039254752764180667221955, −9.039011136761460816234476847665, −7.87509882153338961779475800509, −6.75745450580728103934093269138, −6.08340243692618211787248605447, −4.85977001435120184974922831540, −3.63860602152295520378609640417, −2.62748346072663441462176643778, −2.47013984580750265572655919373,
2.12186676855717505047567588076, 3.48278402689989651239247470701, 4.33088472555526194103687447465, 5.26414385680842896535776859112, 5.88249034581749981724940982997, 7.30396290516018280210614385991, 8.141905218194819398925449230309, 8.843904311396781813859351081080, 9.822198897564259126219635204621, 11.24881496335182295239261310575
