L(s) = 1 | + 5-s + (−0.866 + 0.5i)7-s − i·11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.499 − 0.866i)49-s + (−0.5 − 0.866i)53-s − i·55-s + ⋯ |
L(s) = 1 | + 5-s + (−0.866 + 0.5i)7-s − i·11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.499 − 0.866i)49-s + (−0.5 − 0.866i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.418620655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418620655\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 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 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 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
−8.930054579106656279101457949689, −8.471090716047725031895301165943, −7.38335438218933669118819974048, −6.42547790007052301292026376083, −5.93125340005176059779962328998, −5.47457360067674072990355435016, −4.12976162721740679816401452201, −3.30950899098424419667728438376, −2.40337941388952180740836899914, −1.30117087454715303637527662235,
1.02919977329600778195907393244, 2.30700318763082098220377092855, 3.17377534971714774847937605544, 4.08011534403630535356001535356, 5.23622848318525716016595415803, 5.72985335025634246474894284283, 6.59923345317052854669072940504, 7.36550265378910738641813814531, 7.898739064990568610624159817971, 9.284857819858757035328516028725