L(s) = 1 | − i·3-s − i·7-s − 9-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 21-s − 2i·23-s − 25-s + i·27-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯ |
L(s) = 1 | − i·3-s − i·7-s − 9-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 21-s − 2i·23-s − 25-s + i·27-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8916822029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8916822029\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + 2iT - T^{2} \) |
| 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 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.738347734328461265562415683867, −8.153018245669623429483235877975, −7.49349123923067199855347696382, −6.70891248273647857718847515071, −6.09573983785060478883165724104, −5.05624440779986162453870070605, −4.05971169403785206166445338684, −2.98932195525673536259601004059, −1.95595052725420206781836907020, −0.61648077832895542661799122430,
2.00194846719238086705147570658, 2.96331479897636675510824701269, 3.97926708478563933588316304733, 4.77005920379001538220267151624, 5.67911402293675031802921425337, 6.16285076182293122390775343071, 7.48619052151543352533196767088, 8.167577636947515851017086859124, 9.187289130798607139359461913936, 9.515519978801929088305743294463