L(s) = 1 | + (−0.587 + 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)10-s + i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)10-s + i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9605985573 + 1.153378356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9605985573 + 1.153378356i\) |
\(L(1)\) |
\(\approx\) |
\(0.9200223296 + 0.5781922535i\) |
\(L(1)\) |
\(\approx\) |
\(0.9200223296 + 0.5781922535i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (-0.951 + 0.309i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.951 - 0.309i)T \) |
| 29 | \( 1 + iT \) |
| 31 | \( 1 + (-0.587 - 0.809i)T \) |
| 37 | \( 1 + (-0.587 - 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.951 + 0.309i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.951 - 0.309i)T \) |
| 59 | \( 1 + (0.587 - 0.809i)T \) |
| 67 | \( 1 + (-0.951 + 0.309i)T \) |
| 71 | \( 1 + (0.951 + 0.309i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.951 - 0.309i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.587 - 0.809i)T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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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
−31.34964665157505199977098645748, −30.62808570019724883861211334227, −29.93686075299808910059369538205, −28.92169828787039271890730988907, −27.09076740517131312046037647581, −26.66685714332376400845991516118, −25.63883745997412805998444966424, −24.160990264969102619051694928611, −22.863346187071786667263441728587, −21.40688330015312300555494717696, −20.35121483122237218473046267202, −19.362717146104247032870986110326, −18.45073037555951310978892274512, −17.54499034557236254004745682975, −15.72377923283646963620346143481, −14.00790773858666180632774626140, −13.376596196259625462715244510972, −11.516317920767679075835170915970, −10.76027635678723243017446626444, −9.07032534296070579973307971462, −7.91432422607200669570060720450, −6.87883703929164044981871516220, −3.88330498078853627410070230971, −2.79016874802023989415264285272, −1.02089241963732251659262439122,
1.74142487925873831305144242758, 4.31085857840227942807945091736, 5.447468518343120997306431323465, 7.54805750889493578235339741771, 8.69470352364436128045047708395, 9.29111007670792779188675820251, 10.94605750845168363650931863189, 12.876718213408038383256722991194, 14.3677449273071475913716385242, 15.435517991633923295077054651327, 16.11601436715816141859167269713, 17.55683869120718412623572777191, 18.80589084959483670305537086829, 20.11615183067527987052874936257, 20.8898896662156903189261491836, 22.56331448247042394800774784869, 24.059111791889619539898091142893, 24.94610374349597626512764235215, 25.74300202076205741852007013480, 27.04838662388097532095509397847, 27.916468624749715065414557275003, 28.58625811340366211627066236655, 31.05852231673508282283044768751, 31.416082783097034457662541227210, 32.99234239165709658783276095191