L(s) = 1 | + 1.06i·3-s + (0.727 + 2.24i)5-s + (−0.587 − 0.809i)7-s + 1.85·9-s + (−5.06 − 1.64i)11-s + (−3.77 + 5.19i)13-s + (−2.39 + 0.778i)15-s + (−6.88 − 2.23i)17-s + (−2.95 − 4.06i)19-s + (0.865 − 0.628i)21-s + (−0.323 − 0.235i)23-s + (−0.444 + 0.323i)25-s + 5.19i·27-s + (1.59 − 0.516i)29-s + (0.969 − 2.98i)31-s + ⋯ |
L(s) = 1 | + 0.617i·3-s + (0.325 + 1.00i)5-s + (−0.222 − 0.305i)7-s + 0.618·9-s + (−1.52 − 0.495i)11-s + (−1.04 + 1.44i)13-s + (−0.618 + 0.200i)15-s + (−1.66 − 0.542i)17-s + (−0.677 − 0.933i)19-s + (0.188 − 0.137i)21-s + (−0.0674 − 0.0490i)23-s + (−0.0889 + 0.0646i)25-s + 0.999i·27-s + (0.295 − 0.0959i)29-s + (0.174 − 0.536i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4856691187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4856691187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (-1.37 - 6.25i)T \) |
good | 3 | \( 1 - 1.06iT - 3T^{2} \) |
| 5 | \( 1 + (-0.727 - 2.24i)T + (-4.04 + 2.93i)T^{2} \) |
| 11 | \( 1 + (5.06 + 1.64i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (3.77 - 5.19i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.88 + 2.23i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.95 + 4.06i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.323 + 0.235i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.59 + 0.516i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.969 + 2.98i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 3.75i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-2.21 - 1.61i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (4.48 - 6.16i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.71 + 2.50i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.99 + 6.53i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (10.7 - 7.80i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-12.1 + 3.93i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (10.8 + 3.53i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + (8.48 + 11.6i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.79 - 1.88i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.21926088435951323577424155179, −9.610680827690645043451893979188, −8.826392086056390074897170064010, −7.58580175159840578181981210006, −6.87755060811371319427777962642, −6.27882278831174457862908003066, −4.73636948191129252035376827715, −4.46540324525997889306025342099, −2.91522597835281623631843657296, −2.27044117891342096601633308540,
0.19022103459018541924309614302, 1.82552216562474733401857696593, 2.65608611394415823859804290666, 4.30878338357019753466708850322, 5.11697854779801014682603133435, 5.85185027224100302633065063325, 6.97465328098842240168478730659, 7.77409113129678375307330990445, 8.415232609972085688327317732049, 9.298874201669924621186308834980