L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·8-s − 0.999·10-s + (−0.5 + 0.866i)16-s + 2i·17-s + (−0.866 − 0.499i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)34-s + (−0.499 − 0.866i)40-s + (0.5 + 0.866i)49-s + (0.866 − 0.499i)50-s − 2i·53-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·8-s − 0.999·10-s + (−0.5 + 0.866i)16-s + 2i·17-s + (−0.866 − 0.499i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)34-s + (−0.499 − 0.866i)40-s + (0.5 + 0.866i)49-s + (0.866 − 0.499i)50-s − 2i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.512271517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.512271517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + 2iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.956224672842400579962658046164, −8.572085311240701868569664080843, −8.164791442267288846974001991058, −7.32773550331082389186611026830, −6.54963811539873896149969850818, −5.87940670171164686388235316514, −4.78424812946469498178519331162, −3.91024108092007094341439498897, −3.32885454178301273009373594473, −2.06023936028566122535163056463,
0.925882931185934536573689556123, 2.48602737220232797426061366771, 3.39895092753292895614691590449, 4.36669788582558019045778846385, 4.99548007048139648599869820669, 5.83014395926782614067774759010, 7.04057901194451708312205217036, 7.47412640919528934777026826288, 8.694095520927826334682588040008, 9.403097270269888767349868584639