L(s) = 1 | + (0.473 − 0.880i)2-s + (−0.0448 + 0.998i)3-s + (−0.550 − 0.834i)4-s + (0.309 + 0.951i)5-s + (0.858 + 0.512i)6-s + (0.983 − 0.178i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 − 0.0896i)9-s + (0.983 + 0.178i)10-s + (0.753 + 0.657i)11-s + (0.858 − 0.512i)12-s + (0.753 − 0.657i)13-s + (0.309 − 0.951i)14-s + (−0.963 + 0.266i)15-s + (−0.393 + 0.919i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.473 − 0.880i)2-s + (−0.0448 + 0.998i)3-s + (−0.550 − 0.834i)4-s + (0.309 + 0.951i)5-s + (0.858 + 0.512i)6-s + (0.983 − 0.178i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 − 0.0896i)9-s + (0.983 + 0.178i)10-s + (0.753 + 0.657i)11-s + (0.858 − 0.512i)12-s + (0.753 − 0.657i)13-s + (0.309 − 0.951i)14-s + (−0.963 + 0.266i)15-s + (−0.393 + 0.919i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.184038049 - 0.06291793847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184038049 - 0.06291793847i\) |
\(L(1)\) |
\(\approx\) |
\(1.256906385 - 0.1102192132i\) |
\(L(1)\) |
\(\approx\) |
\(1.256906385 - 0.1102192132i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (0.473 - 0.880i)T \) |
| 3 | \( 1 + (-0.0448 + 0.998i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (0.983 - 0.178i)T \) |
| 11 | \( 1 + (0.753 + 0.657i)T \) |
| 13 | \( 1 + (0.753 - 0.657i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.963 - 0.266i)T \) |
| 23 | \( 1 + (-0.900 - 0.433i)T \) |
| 29 | \( 1 + (0.936 + 0.351i)T \) |
| 31 | \( 1 + (-0.393 - 0.919i)T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.691 - 0.722i)T \) |
| 47 | \( 1 + (-0.0448 - 0.998i)T \) |
| 53 | \( 1 + (-0.550 + 0.834i)T \) |
| 59 | \( 1 + (0.858 - 0.512i)T \) |
| 61 | \( 1 + (0.983 + 0.178i)T \) |
| 67 | \( 1 + (-0.550 - 0.834i)T \) |
| 73 | \( 1 + (0.473 - 0.880i)T \) |
| 79 | \( 1 + (-0.995 + 0.0896i)T \) |
| 83 | \( 1 + (0.858 - 0.512i)T \) |
| 89 | \( 1 + (-0.550 + 0.834i)T \) |
| 97 | \( 1 + (-0.222 - 0.974i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.69431283826966416385281468619, −30.79233366299381664102077419756, −29.87257647831876277484257275594, −28.47293076454629844114652637157, −27.346578884606361179467663460544, −25.80269078591597157699657274918, −24.86612313483257236886528789284, −24.10615792660242463776710055125, −23.51852566435053262353082382575, −21.87633901376806089552725085664, −20.90261226756482144959129345291, −19.35442059890172959253029986341, −17.862907069309452170891139160794, −17.2343734327727143947137491988, −16.05732353212506718422675586261, −14.38550242316569861504260336441, −13.64757508643507893809316696504, −12.499171053502140524542660844955, −11.468579882003709526929574578519, −8.71337936481880986785514199864, −8.34632354842535241143891526058, −6.61908231373028356011517609503, −5.630304659920091628868588556, −4.156065370619570312737105604376, −1.70989835138596392003310845740,
2.20053586064249887424669308267, 3.7355457939131297798431156120, 4.851381429390861092211466649558, 6.35096575576871352750619565844, 8.64774260339287589038661125712, 10.050337935162945312761094349016, 10.86418984826039314533317227190, 11.755530804827259309552291068849, 13.65296048630337976665836938077, 14.62942031102468047216879759200, 15.36765798528748599040816206801, 17.40965618646469640679494932259, 18.259702354975362746041452986577, 19.9127665137097805219738760294, 20.71350729827834263237992234911, 21.80806179385549089961591016156, 22.51057798678947092873283807862, 23.52852126435364822025731558031, 25.25242826560443312374953101853, 26.6241370907837637082972103421, 27.53330608839428553316109544181, 28.28079361026155251640507048989, 29.7838951895367859379323712404, 30.52579225593235007861585147347, 31.50440029043190051676984058922