L(s) = 1 | + 1.41i·2-s + (−0.420 + 1.68i)3-s − 2.00·4-s + 2.16i·5-s + (−2.37 − 0.595i)6-s + 2.64·7-s − 2.82i·8-s + (−2.64 − 1.41i)9-s − 3.06·10-s + (0.841 − 3.36i)12-s − 5.59·13-s + 3.74i·14-s + (−3.64 − 0.913i)15-s + 4.00·16-s + (2 − 3.74i)18-s + 8.66·19-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (−0.242 + 0.970i)3-s − 1.00·4-s + 0.970i·5-s + (−0.970 − 0.242i)6-s + 0.999·7-s − 1.00i·8-s + (−0.881 − 0.471i)9-s − 0.970·10-s + (0.242 − 0.970i)12-s − 1.55·13-s + 1.00i·14-s + (−0.941 − 0.235i)15-s + 1.00·16-s + (0.471 − 0.881i)18-s + 1.98·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117413 + 0.951948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117413 + 0.951948i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (0.420 - 1.68i)T \) |
| 7 | \( 1 - 2.64T \) |
good | 5 | \( 1 - 2.16iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8.66T + 19T^{2} \) |
| 23 | \( 1 - 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.4iT - 59T^{2} \) |
| 61 | \( 1 + 0.543T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 1.40iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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
−13.84706484910960935533535202864, −12.07500226110326982539205972501, −11.18523137887926864502206017870, −9.994456668856462506116965500499, −9.354147311742838844445297523455, −7.86931402009451916537712410624, −7.09046377891745222284176138259, −5.55069630313077150408070311816, −4.82497503698943646906338391506, −3.30789931101140729284184083669,
1.04038786853832952868559517285, 2.52148227824101171287759760405, 4.71011164419727524042511196043, 5.38420495821886226800928382031, 7.39230308092814073859124015808, 8.294238399744542573220714785016, 9.278997095565675602438834019727, 10.54274214622865209348599718580, 11.80022978704059036962540612874, 12.11281352603991876863996768362