| L(s) = 1 | + (−0.0336 − 0.234i)2-s + (0.198 − 0.435i)3-s + (1.86 − 0.547i)4-s + (0.991 − 1.14i)5-s + (−0.108 − 0.0319i)6-s + (2.14 − 1.37i)7-s + (−0.387 − 0.848i)8-s + (1.81 + 2.09i)9-s + (−0.301 − 0.193i)10-s + (−0.479 + 3.33i)11-s + (0.132 − 0.921i)12-s + (2.76 + 1.77i)13-s + (−0.394 − 0.455i)14-s + (−0.301 − 0.659i)15-s + (3.08 − 1.98i)16-s + (−6.02 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.0237 − 0.165i)2-s + (0.114 − 0.251i)3-s + (0.932 − 0.273i)4-s + (0.443 − 0.511i)5-s + (−0.0443 − 0.0130i)6-s + (0.809 − 0.520i)7-s + (−0.136 − 0.299i)8-s + (0.604 + 0.697i)9-s + (−0.0952 − 0.0611i)10-s + (−0.144 + 1.00i)11-s + (0.0382 − 0.266i)12-s + (0.768 + 0.493i)13-s + (−0.105 − 0.121i)14-s + (−0.0777 − 0.170i)15-s + (0.771 − 0.495i)16-s + (−1.46 − 0.428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96876 - 0.780125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96876 - 0.780125i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (0.0336 + 0.234i)T + (-1.91 + 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.198 + 0.435i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.991 + 1.14i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.14 + 1.37i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.479 - 3.33i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 1.77i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (6.02 + 1.76i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (4.05 - 1.19i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (6.03 + 1.77i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.96 + 4.31i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.0248 - 0.0286i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.12 - 2.45i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.535 - 1.17i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + (-5.65 + 3.63i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (8.76 + 5.63i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.49 + 3.27i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.970 + 6.74i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.582 + 4.04i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.18 - 1.22i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (2.02 + 1.29i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.93 - 2.23i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.13 - 11.2i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.95 - 8.02i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.95299407230369589403784385018, −9.976466453609089013935719394834, −9.049651236272621182984479537685, −7.86834424019393778391673929437, −7.16797768640352357069021013367, −6.28716491088701051092208988657, −5.00771912408842519319331623032, −4.14676003844890547658583965327, −2.12911082096237534359692422558, −1.63406472244934884133658806327,
1.80403618315042448803683566033, 2.95098778656251347840279283522, 4.12116294460594248675217274893, 5.66915225965408143490040650118, 6.36381786210541283830585490341, 7.19896878367367920583503357773, 8.502791224319030567891602356083, 8.843559008901119091619183302922, 10.45346481573540170242026766285, 10.83864458499356785910320714631
