L(s) = 1 | + 2.12·2-s + (−19.8 − 18.2i)3-s − 59.4·4-s + (−72.7 + 101. i)5-s + (−42.2 − 38.8i)6-s − 435. i·7-s − 262.·8-s + (60.6 + 726. i)9-s + (−154. + 216. i)10-s − 1.65e3i·11-s + (1.18e3 + 1.08e3i)12-s + 1.64e3i·13-s − 926. i·14-s + (3.30e3 − 689. i)15-s + 3.24e3·16-s − 2.05e3·17-s + ⋯ |
L(s) = 1 | + 0.265·2-s + (−0.735 − 0.677i)3-s − 0.929·4-s + (−0.582 + 0.813i)5-s + (−0.195 − 0.180i)6-s − 1.26i·7-s − 0.512·8-s + (0.0831 + 0.996i)9-s + (−0.154 + 0.216i)10-s − 1.24i·11-s + (0.683 + 0.629i)12-s + 0.747i·13-s − 0.337i·14-s + (0.978 − 0.204i)15-s + 0.792·16-s − 0.417·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0322642 - 0.312575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0322642 - 0.312575i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (19.8 + 18.2i)T \) |
| 5 | \( 1 + (72.7 - 101. i)T \) |
good | 2 | \( 1 - 2.12T + 64T^{2} \) |
| 7 | \( 1 + 435. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.65e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.64e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.05e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 3.63e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.12e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.14e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.34e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 1.77e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 2.29e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.44e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 4.96e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 8.17e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 6.37e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.51e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.04e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.66e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 7.58e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 2.61e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.16e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.20e5iT - 8.32e11T^{2} \) |
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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
−17.54044857104208073514531611640, −16.30339162393016893026330471711, −14.18423546892082179586858020893, −13.44934918157026481534854210183, −11.72309640880427805714239521282, −10.45741544851985405704618206477, −8.011763349663693446892611499749, −6.36795943408830265416563294029, −4.09508038044328891778579596216, −0.23268793191604961560974224557,
4.32448210734120036902524704134, 5.55527682887432286834119884247, 8.564441561128573738316790986821, 9.840231454516263936477964189644, 11.99545411477445886512846122146, 12.70924894734526228316602200055, 14.94399335404258781904021864039, 15.79197649346041549519522996672, 17.40263266174661977727616398330, 18.29885913604611387213621120307