| L(s) = 1 | + (1.75 + 1.54i)2-s + (−1.66 − 0.466i)3-s + (0.439 + 3.45i)4-s + (−3.69 − 0.537i)5-s + (−2.20 − 3.40i)6-s + (−0.274 − 3.03i)7-s + (−1.95 + 2.87i)8-s + (2.56 + 1.55i)9-s + (−5.66 − 6.66i)10-s + (3.06 − 3.11i)11-s + (0.879 − 5.97i)12-s + (−3.11 − 0.112i)13-s + (4.21 − 5.75i)14-s + (5.91 + 2.62i)15-s + (−1.13 + 0.293i)16-s + (4.72 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (1.24 + 1.09i)2-s + (−0.963 − 0.269i)3-s + (0.219 + 1.72i)4-s + (−1.65 − 0.240i)5-s + (−0.901 − 1.38i)6-s + (−0.103 − 1.14i)7-s + (−0.689 + 1.01i)8-s + (0.854 + 0.518i)9-s + (−1.79 − 2.10i)10-s + (0.923 − 0.940i)11-s + (0.253 − 1.72i)12-s + (−0.862 − 0.0311i)13-s + (1.12 − 1.53i)14-s + (1.52 + 0.676i)15-s + (−0.284 + 0.0734i)16-s + (1.14 − 0.530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921807 - 0.444787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921807 - 0.444787i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.66 + 0.466i)T \) |
| 59 | \( 1 + (-1.45 + 7.54i)T \) |
| good | 2 | \( 1 + (-1.75 - 1.54i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (3.69 + 0.537i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.274 + 3.03i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (-3.06 + 3.11i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.11 + 0.112i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-4.72 + 2.18i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (6.56 + 1.44i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (6.38 + 2.41i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (1.18 + 5.90i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (0.603 - 1.90i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (-0.744 - 1.09i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (3.11 + 2.55i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (11.1 - 2.88i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (-3.62 + 0.527i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (3.65 - 4.29i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 4.62i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (1.20 + 2.48i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (-4.75 + 11.9i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (-4.29 - 0.466i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (0.128 - 7.13i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (-2.10 + 1.06i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-2.48 - 0.837i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (2.69 + 3.67i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−11.11561814665724978795726129220, −10.07733401048491018547021339647, −8.243268207979927319253356222964, −7.67678804231583835152041938077, −6.89021767174635344584915477052, −6.22671340945301182787571992609, −4.96203780201185474204187633862, −4.20357055982966098112896168939, −3.62321361797988836382961189876, −0.45398039135286524522886126556,
1.90148123594504035924898030262, 3.51961132322758788962479199085, 4.14221108208603893935435120257, 5.01097992410922490871921701304, 6.00287367797300456590946100706, 7.05931936735117177331403121343, 8.286188230058890348330938489914, 9.730090290424961676768101985735, 10.44525459701428380214558688817, 11.44951877417943374071434309623
