| L(s) = 1 | + (1.98 + 0.436i)2-s + (−0.161 − 0.986i)3-s + (1.92 + 0.891i)4-s + (−0.867 − 3.12i)5-s + (0.109 − 2.02i)6-s + (2.78 + 2.63i)7-s + (0.198 + 0.150i)8-s + (−0.947 + 0.319i)9-s + (−0.356 − 6.57i)10-s + (−3.60 + 4.23i)11-s + (0.567 − 2.04i)12-s + (2.21 + 0.745i)13-s + (4.36 + 6.44i)14-s + (−2.94 + 1.36i)15-s + (−2.42 − 2.85i)16-s + (2.52 − 2.38i)17-s + ⋯ |
| L(s) = 1 | + (1.40 + 0.308i)2-s + (−0.0934 − 0.569i)3-s + (0.963 + 0.445i)4-s + (−0.388 − 1.39i)5-s + (0.0448 − 0.827i)6-s + (1.05 + 0.996i)7-s + (0.0701 + 0.0533i)8-s + (−0.315 + 0.106i)9-s + (−0.112 − 2.07i)10-s + (−1.08 + 1.27i)11-s + (0.163 − 0.590i)12-s + (0.613 + 0.206i)13-s + (1.16 + 1.72i)14-s + (−0.760 + 0.351i)15-s + (−0.605 − 0.712i)16-s + (0.611 − 0.579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08285 - 0.348956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08285 - 0.348956i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.161 + 0.986i)T \) |
| 59 | \( 1 + (-7.48 - 1.74i)T \) |
| good | 2 | \( 1 + (-1.98 - 0.436i)T + (1.81 + 0.839i)T^{2} \) |
| 5 | \( 1 + (0.867 + 3.12i)T + (-4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-2.78 - 2.63i)T + (0.378 + 6.98i)T^{2} \) |
| 11 | \( 1 + (3.60 - 4.23i)T + (-1.77 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 0.745i)T + (10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (-2.52 + 2.38i)T + (0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.59 - 6.50i)T + (-13.7 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-3.27 - 0.356i)T + (22.4 + 4.94i)T^{2} \) |
| 29 | \( 1 + (8.40 - 1.85i)T + (26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (2.83 + 7.12i)T + (-22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-5.49 + 4.18i)T + (9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (0.825 - 0.0897i)T + (40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (0.211 + 0.248i)T + (-6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-1.20 + 4.33i)T + (-40.2 - 24.2i)T^{2} \) |
| 53 | \( 1 + (-0.181 + 3.34i)T + (-52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.100 - 0.0220i)T + (55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (3.91 + 2.97i)T + (17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-0.473 + 1.70i)T + (-60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-2.08 - 3.07i)T + (-27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (1.60 - 9.76i)T + (-74.8 - 25.2i)T^{2} \) |
| 83 | \( 1 + (2.53 - 4.77i)T + (-46.5 - 68.6i)T^{2} \) |
| 89 | \( 1 + (-1.38 + 0.304i)T + (80.7 - 37.3i)T^{2} \) |
| 97 | \( 1 + (3.35 - 4.94i)T + (-35.9 - 90.1i)T^{2} \) |
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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
−12.70404168658807717207398507528, −12.16492308943221793854404011104, −11.31723695813819659587523357452, −9.420563519292045277139586033352, −8.259658998115209110051290955477, −7.43707370805766328023433218331, −5.64697193516904460525361335623, −5.22308804168202827901497325067, −4.12743948725491261725497067821, −2.03731400080622591243754174540,
2.91009657907156057312878665629, 3.74353999624379472576056199962, 4.92864485927015044713054671132, 6.05143993792217773984483081457, 7.35781179298294159433799135893, 8.530902087145863321039635993939, 10.58695492693087678788982182735, 10.98563286171168039164376786220, 11.38084518175184445434692496665, 13.07673042569320542328883262660
