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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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
−10.57740621618608805694527829361, −10.14626422500000299607201316635, −9.041946286245670320815761916392, −8.588684008479695694507575685081, −6.73422212067053265184060490740, −5.93401584933108306776253388564, −4.90056840148405923987728802608, −4.18055981698323905195836329405, −2.88487694441176343880457789316, 0,
1.67691603811137524108392664817, 3.21448565480825621746346424175, 4.83097929236300810904347161634, 6.08473424643705040383204548522, 6.84063791822644202189258750334, 7.33799546108765871606028141286, 8.340447768786625564727949574523, 9.815765458038132012689825511468, 10.61747997355707164025467188021