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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12845 + 0.559408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12845 + 0.559408i\) |
\(L(\frac{3}{2})\) |
|
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
−13.70305202555260878798490505558, −13.24688432492786032185953258752, −11.36073117391426913042391564366, −10.40217630212134984427352541425, −9.412728255827074359655560748809, −8.869829507626480074278667273485, −7.34359416371749824397387048395, −5.47814205782257414762550558126, −4.17284991819335405692382112761, −3.06863409097450051281485096062,
2.01435326820631659756192230674, 3.33607576344146875796464983280, 5.98782144721457735788228088405, 6.79382649383715623852401979701, 8.015935956480590823106124493460, 8.901421381993921158664686369373, 10.16813813781579635199109532636, 11.80082940926999881885742185286, 12.68548751411138965689879934653, 13.34836502465437859034610198566