| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.848117 + 2.08890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.848117 + 2.08890i\) |
| \(L(\frac{3}{2})\) |
|
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
−11.25846100640757245107970804200, −10.89535096294029605067173527270, −9.577554958398064691981205842340, −7.73075194235906176765258638369, −7.08544955789921291382756410389, −6.55931660709102629070027534436, −5.51364473553007325799000426727, −4.94614645647319123045317774593, −3.43126627136536666960502971926, −2.73344355544314738311019382328,
0.893331891629461796850764631480, 2.62126628854938098941769799570, 4.03789329358206240463293732083, 4.84683615377938459131736247648, 5.55380471419827516449295938198, 5.99502663028443055620950997051, 7.72519080821433553395257517325, 8.781220502570310184752523403775, 10.08556298811041582989148691329, 10.85856069692464582391661286867
