L(s) = 1 | + (−0.438 + 0.898i)3-s + (0.961 − 0.275i)4-s + (0.634 − 0.256i)5-s + (−0.615 − 0.788i)9-s + (−0.173 + 0.984i)12-s + (−0.0477 + 0.682i)15-s + (0.848 − 0.529i)16-s + (0.539 − 0.421i)20-s + (0.592 − 1.62i)23-s + (−0.382 + 0.369i)25-s + (0.978 − 0.207i)27-s + (−0.0121 + 0.347i)31-s + (−0.809 − 0.587i)36-s + (1.71 + 0.764i)37-s + (−0.592 − 0.342i)45-s + ⋯ |
L(s) = 1 | + (−0.438 + 0.898i)3-s + (0.961 − 0.275i)4-s + (0.634 − 0.256i)5-s + (−0.615 − 0.788i)9-s + (−0.173 + 0.984i)12-s + (−0.0477 + 0.682i)15-s + (0.848 − 0.529i)16-s + (0.539 − 0.421i)20-s + (0.592 − 1.62i)23-s + (−0.382 + 0.369i)25-s + (0.978 − 0.207i)27-s + (−0.0121 + 0.347i)31-s + (−0.809 − 0.587i)36-s + (1.71 + 0.764i)37-s + (−0.592 − 0.342i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.552723507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.552723507\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.438 - 0.898i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 5 | \( 1 + (-0.634 + 0.256i)T + (0.719 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-0.374 - 0.927i)T^{2} \) |
| 13 | \( 1 + (0.559 - 0.829i)T^{2} \) |
| 17 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 19 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 23 | \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.615 - 0.788i)T^{2} \) |
| 31 | \( 1 + (0.0121 - 0.347i)T + (-0.997 - 0.0697i)T^{2} \) |
| 37 | \( 1 + (-1.71 - 0.764i)T + (0.669 + 0.743i)T^{2} \) |
| 41 | \( 1 + (0.615 + 0.788i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.354 - 1.23i)T + (-0.848 - 0.529i)T^{2} \) |
| 53 | \( 1 + (-1.87 + 0.608i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (1.91 + 0.476i)T + (0.882 + 0.469i)T^{2} \) |
| 61 | \( 1 + (-0.997 + 0.0697i)T^{2} \) |
| 67 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (0.267 + 1.25i)T + (-0.913 + 0.406i)T^{2} \) |
| 73 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 79 | \( 1 + (0.961 - 0.275i)T^{2} \) |
| 83 | \( 1 + (-0.559 - 0.829i)T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.573 - 1.42i)T + (-0.719 - 0.694i)T^{2} \) |
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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
−9.068482012059878077227566650022, −8.181290839232321819884551521764, −7.22795897452659685801915315499, −6.25762507305480360363983571983, −6.01995172488290733601222761834, −5.04681824640938685541060595790, −4.39915802473833887315337234069, −3.18900305423068537436171040498, −2.43551813059524677306271610955, −1.14734180366905901326333585131,
1.29523977070590316367701773510, 2.20077224220236398056444465850, 2.87247539016318517213688786772, 4.02749455130820454830428925568, 5.46779285583475074288790776191, 5.81287686877350844558704114587, 6.62021138596393700660257764007, 7.26326897367712672928808470946, 7.76168207888003584422251889232, 8.622849035272399675504418174262