L(s) = 1 | + (−0.5 + 0.866i)2-s + i·3-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + 7-s − 8-s − 9-s + (−0.866 + 0.499i)10-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + i·3-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + 7-s − 8-s − 9-s + (−0.866 + 0.499i)10-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9436075410\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9436075410\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.71743405253884669359887700806, −9.589883867975464073508482241374, −9.125058152370480146917099174275, −8.358130894526478566837672606986, −7.41958254124965460298007860590, −6.32935663779118177307284738689, −5.78315055244107260913620522363, −4.66297605687337423971570491281, −3.48466624040421063590918278245, −2.23191255081108015681062112832,
1.35327513961044349644508019775, 1.87072151589929830541150941540, 3.06507532298649991736798395559, 4.91803273928049618374612338313, 5.67979925151830447873884987386, 6.67280902556778698565489343659, 7.63376422346620424689575609013, 8.719552738338096698004922850205, 9.145160084580691357198906170668, 10.23789551564847444422593784750