L(s) = 1 | + (1.5 + 2.59i)3-s + (−0.5 + 0.866i)5-s + (2 + 1.73i)7-s + (−3 + 5.19i)9-s + (−0.5 − 0.866i)11-s − 2·13-s − 3·15-s + (−1.5 − 2.59i)17-s + (2.5 − 4.33i)19-s + (−1.5 + 7.79i)21-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s − 9·27-s + 6·29-s + (0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.866 + 1.49i)3-s + (−0.223 + 0.387i)5-s + (0.755 + 0.654i)7-s + (−1 + 1.73i)9-s + (−0.150 − 0.261i)11-s − 0.554·13-s − 0.774·15-s + (−0.363 − 0.630i)17-s + (0.573 − 0.993i)19-s + (−0.327 + 1.70i)21-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s − 1.73·27-s + 1.11·29-s + (0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.996721 + 1.49841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.996721 + 1.49841i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−11.18143546427841463818942759146, −10.42784991413781463946568626589, −9.502098050547633598934452136032, −8.793044937181622240119597585892, −8.080453914350116039418667028901, −6.86698067804866699275782873062, −5.11904549307203993490682241331, −4.75245387324220719512885046039, −3.30848895295155402462994362908, −2.52450582703725952683946258894,
1.13387679500666644814933230384, 2.25056367087603332020954781162, 3.68900193758903036522364230515, 5.02565532107304960916713457428, 6.46639484243756491199304551933, 7.33532063233157414062992868651, 8.031987535371007416867774387683, 8.581922835133778278259751725937, 9.797425823616321587668115275112, 10.92763573543515004069109488814