| L(s) = 1 | + (0.341 + 0.433i)2-s + (0.504 − 1.65i)3-s + (0.394 − 1.64i)4-s + (0.0144 − 0.0792i)5-s + (0.890 − 0.348i)6-s + (−4.36 + 2.41i)7-s + (1.85 − 0.856i)8-s + (−2.49 − 1.67i)9-s + (0.0393 − 0.0208i)10-s + (−4.51 − 2.94i)11-s + (−2.53 − 1.48i)12-s + (2.43 − 2.22i)13-s + (−2.53 − 1.06i)14-s + (−0.124 − 0.0639i)15-s + (−2.01 − 1.02i)16-s + (0.996 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.306i)2-s + (0.291 − 0.956i)3-s + (0.197 − 0.824i)4-s + (0.00647 − 0.0354i)5-s + (0.363 − 0.142i)6-s + (−1.64 + 0.912i)7-s + (0.654 − 0.302i)8-s + (−0.830 − 0.556i)9-s + (0.0124 − 0.00658i)10-s + (−1.36 − 0.886i)11-s + (−0.731 − 0.428i)12-s + (0.675 − 0.616i)13-s + (−0.678 − 0.284i)14-s + (−0.0320 − 0.0165i)15-s + (−0.504 − 0.256i)16-s + (0.241 + 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.440116 - 1.08447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.440116 - 1.08447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.504 + 1.65i)T \) |
| 59 | \( 1 + (-7.58 - 1.23i)T \) |
| good | 2 | \( 1 + (-0.341 - 0.433i)T + (-0.465 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.0144 + 0.0792i)T + (-4.67 - 1.76i)T^{2} \) |
| 7 | \( 1 + (4.36 - 2.41i)T + (3.71 - 5.93i)T^{2} \) |
| 11 | \( 1 + (4.51 + 2.94i)T + (4.43 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.43 + 2.22i)T + (1.17 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.996 - 1.65i)T + (-7.96 + 15.0i)T^{2} \) |
| 19 | \( 1 + (-0.988 + 3.55i)T + (-16.2 - 9.79i)T^{2} \) |
| 23 | \( 1 + (-0.781 + 1.61i)T + (-14.2 - 18.0i)T^{2} \) |
| 29 | \( 1 + (0.436 - 3.00i)T + (-27.7 - 8.26i)T^{2} \) |
| 31 | \( 1 + (6.19 + 6.08i)T + (0.559 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-3.49 + 7.56i)T + (-23.9 - 28.1i)T^{2} \) |
| 41 | \( 1 + (0.841 - 11.6i)T + (-40.5 - 5.90i)T^{2} \) |
| 43 | \( 1 + (1.03 - 2.03i)T + (-25.4 - 34.6i)T^{2} \) |
| 47 | \( 1 + (-8.36 + 1.52i)T + (43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (0.121 + 0.0645i)T + (29.7 + 43.8i)T^{2} \) |
| 61 | \( 1 + (1.93 - 1.52i)T + (14.1 - 59.3i)T^{2} \) |
| 67 | \( 1 + (-5.73 + 8.13i)T + (-22.5 - 63.0i)T^{2} \) |
| 71 | \( 1 + (-2.60 + 2.21i)T + (11.4 - 70.0i)T^{2} \) |
| 73 | \( 1 + (0.867 + 1.14i)T + (-19.5 + 70.3i)T^{2} \) |
| 79 | \( 1 + (-2.78 + 6.31i)T + (-53.2 - 58.3i)T^{2} \) |
| 83 | \( 1 + (0.0636 - 0.316i)T + (-76.5 - 32.1i)T^{2} \) |
| 89 | \( 1 + (4.12 + 10.3i)T + (-64.6 + 61.2i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 4.48i)T + (-67.9 + 69.2i)T^{2} \) |
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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
−10.59374963646691769915290559507, −9.467946099859176408882330561537, −8.756859139276044035846753991607, −7.65999927109352493119476482111, −6.65672033023099383205188559886, −5.89980664013406967223568133550, −5.42656095281342844690467924796, −3.29174948470954853884988364105, −2.48507016685568467029506787704, −0.57638533557805760608438543418,
2.56572536832856749622121581360, 3.47600879083962721390224665975, 4.14735071906211812213844596573, 5.35763575485033211324919510782, 6.81300331397251377577766111479, 7.53770905816749386606371841721, 8.623379732274219227460871395014, 9.633947348515893654786314318349, 10.33921117951695686152638553248, 10.89857292489519174690453107521
