| L(s) = 1 | + (0.845 − 0.186i)2-s + (−0.0436 + 1.73i)3-s + (−1.13 + 0.525i)4-s + (1.97 + 0.547i)5-s + (0.285 + 1.47i)6-s + (0.395 − 0.374i)7-s + (−2.24 + 1.70i)8-s + (−2.99 − 0.151i)9-s + (1.76 + 0.0959i)10-s + (2.26 + 2.66i)11-s + (−0.859 − 1.98i)12-s + (−0.338 − 1.00i)13-s + (0.264 − 0.390i)14-s + (−1.03 + 3.39i)15-s + (0.0419 − 0.0493i)16-s + (2.83 − 2.99i)17-s + ⋯ |
| L(s) = 1 | + (0.597 − 0.131i)2-s + (−0.0251 + 0.999i)3-s + (−0.567 + 0.262i)4-s + (0.882 + 0.244i)5-s + (0.116 + 0.600i)6-s + (0.149 − 0.141i)7-s + (−0.792 + 0.602i)8-s + (−0.998 − 0.0503i)9-s + (0.559 + 0.0303i)10-s + (0.682 + 0.803i)11-s + (−0.248 − 0.573i)12-s + (−0.0937 − 0.278i)13-s + (0.0707 − 0.104i)14-s + (−0.267 + 0.875i)15-s + (0.0104 − 0.0123i)16-s + (0.688 − 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22874 + 0.797934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22874 + 0.797934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0436 - 1.73i)T \) |
| 59 | \( 1 + (-7.14 + 2.81i)T \) |
| good | 2 | \( 1 + (-0.845 + 0.186i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-1.97 - 0.547i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.395 + 0.374i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.26 - 2.66i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.338 + 1.00i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (-2.83 + 2.99i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.103 + 0.259i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-4.94 + 0.537i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-1.72 + 7.83i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (7.50 + 2.99i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.225 + 0.296i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (0.267 - 2.46i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-5.32 - 4.52i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-1.24 - 4.49i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (12.2 - 0.665i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (0.156 + 0.710i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (6.26 + 8.24i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-0.609 + 0.169i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (9.03 + 6.12i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 12.0i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-5.88 - 11.1i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-3.67 - 0.809i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (0.834 - 0.565i)T + (35.9 - 90.1i)T^{2} \) |
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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
−12.97073064167497110714948256631, −11.93730584375103261321241657669, −10.91474397138890704766736667545, −9.553045291985427756483232528565, −9.408840710156192661515297986677, −7.84926936174916129135696674768, −6.12853762807859592169926995589, −5.09648883021416296133472334877, −4.13703896079872666224095969477, −2.78231054057478893973701472172,
1.43737620455819403098270572491, 3.44024856672264420922919662771, 5.27560131739951677985505694118, 5.92711367256134929551455987365, 7.01256347479992965649791551572, 8.634278818555885840329345646239, 9.196638838650215219261412683673, 10.61717914059059733088611694644, 11.90222403364243590248510471189, 12.80815567959319900067075635496
