L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − i·23-s − i·27-s + 29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − i·23-s − i·27-s + 29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.031261105 - 0.2125236728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.031261105 - 0.2125236728i\) |
\(L(1)\) |
\(\approx\) |
\(0.8120581955 - 0.1285576488i\) |
\(L(1)\) |
\(\approx\) |
\(0.8120581955 - 0.1285576488i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 - iT \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.68504716279966308141478842723, −20.0803472141893353310468690125, −19.13139369204089921374784314976, −18.345896963925319108356940341883, −17.68105161518033368117049240963, −16.92191580178823286374133861952, −16.01598753546034020669799879572, −15.76062358291886656640917187169, −14.86735650696825232623105375185, −13.8238823622803368028469296631, −13.00565758522190461399866130051, −12.12314534184508485238398294636, −11.63122324177131854793581396139, −10.75367235374198006247084452539, −9.92599108902021011791366528708, −9.2539647143874032611566999527, −8.54888434240294969223219101199, −7.14978514601475169568052043790, −6.471862807340297778999937144206, −5.76049606707156395351711564518, −5.04993970955494483199781922542, −3.74501473264900013717221078344, −3.46155376747728848243278859997, −1.915099694796826047168180314904, −0.70737273374563852512851100447,
0.75450080980258456930415400403, 1.63452935199848842335803946598, 2.88547081949781236024536210635, 4.04202451163798731424490802094, 4.659356047995538802032975503624, 5.93434412292809960679328290462, 6.63415872909330877050055824942, 6.882175293712051435947424291, 8.14212292722580111872407801828, 9.0745218254960585271725935965, 9.87249218614622966619468719777, 10.93117533182295784463146024608, 11.275584047973134988223220880905, 12.30457581033756476503162380152, 12.95579422435229710795873370036, 13.62299677629702016472754932689, 14.38713770282840725887628172095, 15.62552001794742187900509524005, 16.245383459298988646723752225320, 16.84300998835423306155063615512, 17.60539552224366338587547053021, 18.27779314277756125739052341943, 19.15609198031006609316543061751, 19.71371161503574728367347738188, 20.44002542876167216378752147858