L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)4-s + i·5-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.499 − 0.866i)12-s + (0.5 − 0.866i)15-s + (0.499 + 0.866i)16-s + (−0.5 + 0.866i)20-s + (1.36 − 0.366i)23-s − 25-s − 0.999i·27-s + (0.866 − 1.5i)31-s + 0.999·33-s + 0.999i·36-s + (0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)4-s + i·5-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.499 − 0.866i)12-s + (0.5 − 0.866i)15-s + (0.499 + 0.866i)16-s + (−0.5 + 0.866i)20-s + (1.36 − 0.366i)23-s − 25-s − 0.999i·27-s + (0.866 − 1.5i)31-s + 0.999·33-s + 0.999i·36-s + (0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7909000775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7909000775\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−11.32359911362484093318875105356, −10.61823281822263638790643081830, −9.892269155656133594933242152304, −8.138274389718141439876325593246, −7.44442410498619269434109463450, −6.72408961889260864726577211928, −5.99523151051759443651977132911, −4.71582593023957418227651834504, −3.09958121648173750231715939078, −2.08029747378175582595134055297,
1.23492128669739442889899440459, 3.10598679305666000735317067272, 4.73364441736028682641559194240, 5.36265622384288848972000656671, 6.23260338478980074388548502978, 7.26303785786482097245972056960, 8.449927362760735828764032629997, 9.499936602162088523128288025302, 10.31255438674692668626928065955, 11.07296355807486084267396404534