| L(s) = 1 | + (−0.465 − 0.327i)2-s + (−1.65 + 0.518i)3-s + (−0.563 − 1.57i)4-s + (0.110 − 0.0610i)5-s + (0.939 + 0.300i)6-s + (−0.553 + 0.280i)7-s + (−0.559 + 2.01i)8-s + (2.46 − 1.71i)9-s + (−0.0713 − 0.00775i)10-s + (−0.0923 + 0.0348i)11-s + (1.75 + 2.31i)12-s + (−0.725 − 5.70i)13-s + (0.349 + 0.0508i)14-s + (−0.150 + 0.158i)15-s + (−1.67 + 1.36i)16-s + (−0.0507 − 0.936i)17-s + ⋯ |
| L(s) = 1 | + (−0.329 − 0.231i)2-s + (−0.954 + 0.299i)3-s + (−0.281 − 0.789i)4-s + (0.0493 − 0.0273i)5-s + (0.383 + 0.122i)6-s + (−0.209 + 0.106i)7-s + (−0.197 + 0.712i)8-s + (0.820 − 0.571i)9-s + (−0.0225 − 0.00245i)10-s + (−0.0278 + 0.0105i)11-s + (0.505 + 0.668i)12-s + (−0.201 − 1.58i)13-s + (0.0933 + 0.0135i)14-s + (−0.0388 + 0.0408i)15-s + (−0.417 + 0.342i)16-s + (−0.0123 − 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0342077 + 0.0737189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0342077 + 0.0737189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 - 0.518i)T \) |
| 59 | \( 1 + (-4.58 - 6.16i)T \) |
| good | 2 | \( 1 + (0.465 + 0.327i)T + (0.672 + 1.88i)T^{2} \) |
| 5 | \( 1 + (-0.110 + 0.0610i)T + (2.65 - 4.23i)T^{2} \) |
| 7 | \( 1 + (0.553 - 0.280i)T + (4.13 - 5.64i)T^{2} \) |
| 11 | \( 1 + (0.0923 - 0.0348i)T + (8.25 - 7.27i)T^{2} \) |
| 13 | \( 1 + (0.725 + 5.70i)T + (-12.5 + 3.25i)T^{2} \) |
| 17 | \( 1 + (0.0507 + 0.936i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (3.59 - 3.40i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (0.737 - 2.32i)T + (-18.8 - 13.2i)T^{2} \) |
| 29 | \( 1 + (-0.651 - 7.19i)T + (-28.5 + 5.20i)T^{2} \) |
| 31 | \( 1 + (3.42 - 1.01i)T + (25.9 - 16.9i)T^{2} \) |
| 37 | \( 1 + (0.0866 + 0.312i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (1.49 - 1.63i)T + (-3.69 - 40.8i)T^{2} \) |
| 43 | \( 1 + (4.16 - 3.41i)T + (8.48 - 42.1i)T^{2} \) |
| 47 | \( 1 + (9.41 + 5.21i)T + (24.9 + 39.8i)T^{2} \) |
| 53 | \( 1 + (-7.73 + 0.841i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (8.49 + 5.98i)T + (20.5 + 57.4i)T^{2} \) |
| 67 | \( 1 + (3.51 + 3.57i)T + (-1.20 + 66.9i)T^{2} \) |
| 71 | \( 1 + (12.6 - 7.62i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (3.50 + 8.79i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (0.893 + 0.787i)T + (9.95 + 78.3i)T^{2} \) |
| 83 | \( 1 + (-16.5 - 1.19i)T + (82.1 + 11.9i)T^{2} \) |
| 89 | \( 1 + (-4.92 - 2.28i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-4.17 + 0.607i)T + (92.9 - 27.6i)T^{2} \) |
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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
−10.89896820140340797566160063627, −10.32601973427561667689093525924, −9.709488862118881638874125397523, −8.739820935310353278134673547153, −7.56751353916630905369386523731, −6.30893363842620946213475993547, −5.54800423348963146416516336216, −4.87470405121493318705685750962, −3.40265158220312139855096165768, −1.51874906564853602005426277423,
0.06041697484853265192833004453, 2.17924102535287255015836832927, 4.00642962561445978453909601228, 4.70860451865974355581834057729, 6.27198752975689625532347512041, 6.78017879122579966272533928845, 7.72644510661771664139706898523, 8.706468534006272227905632513654, 9.617028251348322008124944758684, 10.50397737926410200654013403695
