L(s) = 1 | + (0.866 − 0.5i)2-s + (0.559 + 0.323i)3-s + (0.499 − 0.866i)4-s + (−2.68 + 1.54i)5-s + 0.646·6-s − 0.999i·8-s + (−1.29 − 2.23i)9-s + (−1.54 + 2.68i)10-s + (−1.52 − 2.94i)11-s + (0.559 − 0.323i)12-s + 0.646·13-s − 2·15-s + (−0.5 − 0.866i)16-s + (1.87 − 3.24i)17-s + (−2.23 − 1.29i)18-s + (0.901 + 1.56i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.323 + 0.186i)3-s + (0.249 − 0.433i)4-s + (−1.19 + 0.692i)5-s + 0.263·6-s − 0.353i·8-s + (−0.430 − 0.745i)9-s + (−0.489 + 0.847i)10-s + (−0.458 − 0.888i)11-s + (0.161 − 0.0932i)12-s + 0.179·13-s − 0.516·15-s + (−0.125 − 0.216i)16-s + (0.453 − 0.785i)17-s + (−0.527 − 0.304i)18-s + (0.206 + 0.358i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.250773514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250773514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.52 + 2.94i)T \) |
good | 3 | \( 1 + (-0.559 - 0.323i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.68 - 1.54i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 0.646T + 13T^{2} \) |
| 17 | \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.901 - 1.56i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.58iT - 29T^{2} \) |
| 31 | \( 1 + (7.48 + 4.32i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.79 + 3.10i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 + (-8.60 + 4.96i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.79 + 10.0i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.559 + 0.323i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.969 + 1.67i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.79 + 6.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (8.06 - 13.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + (8.48 - 4.89i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−9.727393987058334633138952377419, −8.698071200040422634994926957602, −7.932464687094492733015089997590, −7.07202263054775400988307237177, −6.12341870599821779444023317500, −5.20582518021398750756475879569, −3.82984744883634295931882053584, −3.50111370657187049043611109626, −2.52540698437774918240218203931, −0.42170164936454081909476076403,
1.79178438191096452732640328307, 3.18707279263916645296937478979, 4.08129037289889954343897933154, 4.98145656003513615490467361316, 5.66867955746765101827398957151, 7.14850480469811180697286709217, 7.58338761385845302627372606759, 8.358429634154407820378805534131, 8.973565218650888686137447651996, 10.31876510158348294426915628997