L(s) = 1 | + i·3-s + (−0.866 − 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−1.36 + 1.36i)19-s + (−0.866 + 0.5i)25-s − i·27-s + (−1.36 − 0.366i)31-s + (0.866 + 0.5i)36-s + (0.5 + 0.133i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.499i)48-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.866 − 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−1.36 + 1.36i)19-s + (−0.866 + 0.5i)25-s − i·27-s + (−1.36 − 0.366i)31-s + (0.866 + 0.5i)36-s + (0.5 + 0.133i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.499i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2065742741\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2065742741\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + iT - T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.762952571265420112150601943243, −9.245222113720147819105767276661, −8.337051700387611577796610593061, −7.77485456718692723425788698450, −6.32002768437934809944348336284, −5.63125842014480916359814700242, −4.91462636011125901705060056737, −4.12102083222920246905467550340, −3.38228421077878209110660240298, −1.94045938752911281675430471050,
0.14485906872427889621092487023, 1.94568860600968067248310252991, 2.89103132590686388915593419621, 4.07576047968334170661709160599, 4.89760807258971427383140061147, 5.81973955234642667384501560513, 6.88005798180805097150728296988, 7.35056009940138684526931953451, 8.295884762509138737817817846094, 8.829375789260086540890173838947