L(s) = 1 | + (−1.5 + 2.59i)3-s + (2 + 3.46i)5-s + (2.5 + 0.866i)7-s + (−3 − 5.19i)9-s + (−0.5 + 0.866i)11-s − 13-s − 12·15-s + (−1 + 1.73i)17-s + (3 + 5.19i)19-s + (−6 + 5.19i)21-s + (−1 − 1.73i)23-s + (−5.49 + 9.52i)25-s + 9·27-s + 29-s + (2 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.49i)3-s + (0.894 + 1.54i)5-s + (0.944 + 0.327i)7-s + (−1 − 1.73i)9-s + (−0.150 + 0.261i)11-s − 0.277·13-s − 3.09·15-s + (−0.242 + 0.420i)17-s + (0.688 + 1.19i)19-s + (−1.30 + 1.13i)21-s + (−0.208 − 0.361i)23-s + (−1.09 + 1.90i)25-s + 1.73·27-s + 0.185·29-s + (0.359 − 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.397971650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397971650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.5 + 12.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−10.08638981230815411252186910381, −9.882296990117661441077467419163, −8.734944525747083826363749798410, −7.62170568285111875837091377230, −6.52001608261861316084556776059, −5.81673251503624185953993677645, −5.18526133747664355758161892520, −4.17910340465460651190994476003, −3.18201034635913083907509214112, −2.01069426022908631418732977729,
0.70056852305734365499761284713, 1.42310065392705950443276634174, 2.41263048848555210858096599113, 4.59396525793796595078759377340, 5.21753979419448129231833051443, 5.78049644404974675364634348466, 6.84535999236383844040241066768, 7.57434346401167158391202030178, 8.412176934509938844877877507540, 9.067749246381431336561491790921