L(s) = 1 | + (−0.535 − 0.844i)3-s + (−0.0627 + 0.998i)5-s + (0.535 + 0.844i)7-s + (−0.425 + 0.904i)9-s + (0.809 − 0.587i)11-s + (0.187 − 0.982i)13-s + (0.876 − 0.481i)15-s + (0.637 + 0.770i)17-s + (0.728 − 0.684i)19-s + (0.425 − 0.904i)21-s + (−0.728 − 0.684i)23-s + (−0.992 − 0.125i)25-s + (0.992 − 0.125i)27-s + (−0.425 + 0.904i)29-s + (−0.425 + 0.904i)31-s + ⋯ |
L(s) = 1 | + (−0.535 − 0.844i)3-s + (−0.0627 + 0.998i)5-s + (0.535 + 0.844i)7-s + (−0.425 + 0.904i)9-s + (0.809 − 0.587i)11-s + (0.187 − 0.982i)13-s + (0.876 − 0.481i)15-s + (0.637 + 0.770i)17-s + (0.728 − 0.684i)19-s + (0.425 − 0.904i)21-s + (−0.728 − 0.684i)23-s + (−0.992 − 0.125i)25-s + (0.992 − 0.125i)27-s + (−0.425 + 0.904i)29-s + (−0.425 + 0.904i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.573088352 + 0.5729812879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573088352 + 0.5729812879i\) |
\(L(1)\) |
\(\approx\) |
\(1.032621432 + 0.04379094869i\) |
\(L(1)\) |
\(\approx\) |
\(1.032621432 + 0.04379094869i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (-0.535 - 0.844i)T \) |
| 5 | \( 1 + (-0.0627 + 0.998i)T \) |
| 7 | \( 1 + (0.535 + 0.844i)T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.187 - 0.982i)T \) |
| 17 | \( 1 + (0.637 + 0.770i)T \) |
| 19 | \( 1 + (0.728 - 0.684i)T \) |
| 23 | \( 1 + (-0.728 - 0.684i)T \) |
| 29 | \( 1 + (-0.425 + 0.904i)T \) |
| 31 | \( 1 + (-0.425 + 0.904i)T \) |
| 37 | \( 1 + (0.992 + 0.125i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.0627 + 0.998i)T \) |
| 47 | \( 1 + (-0.0627 + 0.998i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.968 - 0.248i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.425 - 0.904i)T \) |
| 71 | \( 1 + (-0.876 - 0.481i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.728 + 0.684i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.0627 - 0.998i)T \) |
| 97 | \( 1 + (0.535 - 0.844i)T \) |
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
−17.4327739998870932208077545445, −16.90007132459923002988037484143, −16.27371851854495750563774004042, −16.10324799390893007719764621440, −14.93444728871668719584656815662, −14.49223319980114293259679046548, −13.749478524117331787794191101237, −13.10881392864457562852898817187, −11.89719219771423889666947934825, −11.82362166848474034915348798977, −11.26209240931083855775486815330, −10.10773098134295037182379939133, −9.681339727503983077261439854929, −9.2489361695017829603402527844, −8.32815503952695637254272749702, −7.56813386671748720229886586130, −6.90788089839066163795109213228, −5.84154071337624362122661815443, −5.40599052062551525187985271019, −4.51286409711020176376806845195, −4.02759437911856828687207312764, −3.674656249734569782833506051085, −2.14523434011535551901154915217, −1.31337895763061866347103314883, −0.58562540252911287127366018589,
0.893290856296486345086299052270, 1.60496187843371406855263688269, 2.577601176991994334403907468622, 3.06655821279377566437649439559, 4.0144109506480222847025356061, 5.13284246703460904123464396063, 5.81979963841945131659903130214, 6.15965790453837592216086858560, 6.99726667009658110298175125104, 7.699429871611599346746975542581, 8.25238189040583470283667272486, 8.97686039195905924158846257000, 9.95476982028522580747094066885, 10.88663686028669876652169079155, 11.072911898688449560802122451662, 11.89207733065660383868537086064, 12.405341222780839156133478474838, 13.09111171536498004585295128651, 13.97433224795637288677380109131, 14.526171064036913718172371802032, 14.92937562653726470802639792669, 15.96827217159522006559767889706, 16.417145748793565565250653527833, 17.53620692805554693154357194732, 17.765115140628658583654529010402