| L(s) = 1 | + (−1.75 − 1.54i)2-s + (−1.06 − 1.36i)3-s + (0.436 + 3.43i)4-s + (3.38 + 0.493i)5-s + (−0.253 + 4.03i)6-s + (−0.172 − 1.90i)7-s + (1.91 − 2.82i)8-s + (−0.746 + 2.90i)9-s + (−5.18 − 6.10i)10-s + (−3.75 + 3.81i)11-s + (4.23 − 4.24i)12-s + (3.27 + 0.118i)13-s + (−2.64 + 3.61i)14-s + (−2.92 − 5.16i)15-s + (−1.02 + 0.265i)16-s + (5.38 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 1.09i)2-s + (−0.612 − 0.790i)3-s + (0.218 + 1.71i)4-s + (1.51 + 0.220i)5-s + (−0.103 + 1.64i)6-s + (−0.0652 − 0.720i)7-s + (0.677 − 0.999i)8-s + (−0.248 + 0.968i)9-s + (−1.63 − 1.92i)10-s + (−1.13 + 1.15i)11-s + (1.22 − 1.22i)12-s + (0.909 + 0.0328i)13-s + (−0.706 + 0.965i)14-s + (−0.754 − 1.33i)15-s + (−0.256 + 0.0663i)16-s + (1.30 − 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.527734 - 0.620356i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.527734 - 0.620356i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.06 + 1.36i)T \) |
| 59 | \( 1 + (1.93 + 7.43i)T \) |
| good | 2 | \( 1 + (1.75 + 1.54i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (-3.38 - 0.493i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.172 + 1.90i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (3.75 - 3.81i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.27 - 0.118i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-5.38 + 2.49i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-6.44 - 1.41i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (1.65 + 0.623i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (-0.308 - 1.53i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (0.0975 - 0.307i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (2.49 + 3.68i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (-3.98 - 3.26i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 2.82i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (6.93 - 1.00i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (4.89 - 5.76i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (5.37 + 4.73i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (3.50 + 7.22i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 3.88i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (9.42 + 1.02i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (0.0945 - 5.23i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (0.872 - 0.442i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 3.38i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (-9.42 - 12.8i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−10.44745780719264426260034884258, −9.964517911848496688340983003297, −9.243484615385850165654741173978, −7.76550045964595257598862224631, −7.41602265816390380397898025878, −6.04909283062804952078719390623, −5.15699207299757269423128231449, −3.06834675378758325376825137190, −1.97964142394276081946801795897, −1.05960387242408141647530064933,
1.09611978128141688610323081505, 3.12126624200169786600447409792, 5.29681444420177175647569439656, 5.83585472409346551957234937992, 6.09270403478707293288551070966, 7.65716457666042747272079366964, 8.681825497361319211511769323807, 9.222862787987037218626590421576, 10.07377213058753037566171173624, 10.47841601585082351480430570518
