L(s) = 1 | + (1.69 + 1.49i)2-s + (−0.860 − 1.50i)3-s + (0.390 + 3.07i)4-s + (2.14 + 0.311i)5-s + (0.786 − 3.83i)6-s + (−0.283 − 3.12i)7-s + (−1.39 + 2.05i)8-s + (−1.52 + 2.58i)9-s + (3.16 + 3.73i)10-s + (1.92 − 1.96i)11-s + (4.28 − 3.23i)12-s + (3.28 + 0.118i)13-s + (4.19 − 5.72i)14-s + (−1.37 − 3.49i)15-s + (0.576 − 0.148i)16-s + (−4.21 + 1.94i)17-s + ⋯ |
L(s) = 1 | + (1.19 + 1.05i)2-s + (−0.496 − 0.867i)3-s + (0.195 + 1.53i)4-s + (0.959 + 0.139i)5-s + (0.321 − 1.56i)6-s + (−0.107 − 1.18i)7-s + (−0.492 + 0.725i)8-s + (−0.506 + 0.862i)9-s + (1.00 + 1.17i)10-s + (0.581 − 0.592i)11-s + (1.23 − 0.932i)12-s + (0.910 + 0.0328i)13-s + (1.12 − 1.53i)14-s + (−0.355 − 0.901i)15-s + (0.144 − 0.0372i)16-s + (−1.02 + 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.64514 + 0.515882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.64514 + 0.515882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.860 + 1.50i)T \) |
| 59 | \( 1 + (3.62 - 6.77i)T \) |
good | 2 | \( 1 + (-1.69 - 1.49i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (-2.14 - 0.311i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.283 + 3.12i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 1.96i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.28 - 0.118i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (4.21 - 1.94i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-1.47 - 0.324i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (1.53 + 0.581i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (-1.68 - 8.39i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (-0.396 + 1.25i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 3.00i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (5.28 + 4.32i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (1.04 - 0.270i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (8.85 - 1.28i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (1.00 - 1.17i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (6.02 + 5.30i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (-1.87 - 3.86i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (-2.50 + 6.28i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (11.9 + 1.29i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-0.122 + 6.76i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (8.67 - 4.40i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-14.9 - 5.02i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (8.77 + 11.9i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−11.01642799797210738615220549367, −10.27036114210072201592980865399, −8.774063677342411131358383239479, −7.77012799820109171496791372593, −6.72406079843395169478091211867, −6.41926707274106199723518445397, −5.60667122110656228483667024682, −4.49051589743375939139473291463, −3.34507676464793930119155953370, −1.45076954806313660242268854198,
1.82480465373388079388508293685, 2.92215360966673959075993359165, 4.12952096482339906139654858849, 4.98240351142840202570039792611, 5.84564073629934292869053297654, 6.38313351813819811134217803344, 8.557066447299533946573516437826, 9.529631899858483935469660971230, 9.955103863125159439675833828223, 11.12650134267916321849457313371