L(s) = 1 | + (−0.669 − 0.743i)3-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.809 − 0.587i)13-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (0.104 − 0.994i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s + 43-s + ⋯ |
L(s) = 1 | + (−0.669 − 0.743i)3-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.809 − 0.587i)13-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (0.104 − 0.994i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08861197595 + 0.1273482197i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08861197595 + 0.1273482197i\) |
\(L(1)\) |
\(\approx\) |
\(0.6452904191 - 0.1565699485i\) |
\(L(1)\) |
\(\approx\) |
\(0.6452904191 - 0.1565699485i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.913 - 0.406i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (0.104 - 0.994i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.978 + 0.207i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.978 - 0.207i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.104 + 0.994i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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
−20.75650674376827550291792130459, −19.88924201718755409417742983128, −19.09521321620833101513471668812, −18.08968343003477477705692150973, −17.444106099023190124264718720184, −16.82171639736662224977873794776, −16.0815668875796637829986092177, −15.23513415329602392309653277487, −14.71216883709565836137844845803, −13.81698060973269135342107163789, −12.58907305224685239454498418100, −12.13263473801616015957530017386, −11.304723333354310883839866163410, −10.4810918709576311655664920335, −9.62013696880487448849635452112, −9.321726078080320564557325302032, −7.9471098627040632023322709690, −7.13128850439899089934208511993, −6.25197263031724794538319085241, −5.344509662042581754163130121721, −4.58378091552321352067062664384, −3.92501573112017928227478585800, −2.73166411893236640591323277358, −1.64754615252225872210739253211, −0.06892898293898049890779482156,
1.14105187312548180677750752315, 2.173379432862509395643839651064, 3.18682557444106837370802055981, 4.29191336059767812017125445427, 5.55855278542337377253586945707, 5.77853937155781898205481140872, 6.84675794658486081986949900236, 7.792227933409738599379209959163, 8.21298511815584045959519102635, 9.44018530388924393188145624240, 10.47356027740437567528512531876, 10.95424740133198895976604961916, 11.96005834561413965347955307201, 12.60224734201810994578027802284, 13.13200998755381540399870978198, 14.33651507412801273891204773247, 14.608612968398671140962945159018, 16.18080985211062862487116160490, 16.39955425993101669624263673496, 17.31432746971615420670578863775, 18.02904712297956481654439344665, 18.76059919627217481387064475823, 19.3508975389805828943843850193, 20.10480271002414512294094288070, 21.18680863575110265705152943706