L(s) = 1 | + (0.783 + 0.620i)2-s + (−0.984 + 0.174i)3-s + (0.229 + 0.973i)4-s + (0.809 + 0.586i)5-s + (−0.880 − 0.474i)6-s + (−0.818 − 0.573i)7-s + (−0.424 + 0.905i)8-s + (0.939 − 0.343i)9-s + (0.270 + 0.962i)10-s + (−0.954 + 0.298i)11-s + (−0.395 − 0.918i)12-s + (0.486 − 0.873i)13-s + (−0.285 − 0.958i)14-s + (−0.899 − 0.436i)15-s + (−0.894 + 0.446i)16-s + (0.690 − 0.722i)17-s + ⋯ |
L(s) = 1 | + (0.783 + 0.620i)2-s + (−0.984 + 0.174i)3-s + (0.229 + 0.973i)4-s + (0.809 + 0.586i)5-s + (−0.880 − 0.474i)6-s + (−0.818 − 0.573i)7-s + (−0.424 + 0.905i)8-s + (0.939 − 0.343i)9-s + (0.270 + 0.962i)10-s + (−0.954 + 0.298i)11-s + (−0.395 − 0.918i)12-s + (0.486 − 0.873i)13-s + (−0.285 − 0.958i)14-s + (−0.899 − 0.436i)15-s + (−0.894 + 0.446i)16-s + (0.690 − 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1529448786 - 0.1478893000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1529448786 - 0.1478893000i\) |
\(L(1)\) |
\(\approx\) |
\(0.9091466199 + 0.4522449907i\) |
\(L(1)\) |
\(\approx\) |
\(0.9091466199 + 0.4522449907i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (0.783 + 0.620i)T \) |
| 3 | \( 1 + (-0.984 + 0.174i)T \) |
| 5 | \( 1 + (0.809 + 0.586i)T \) |
| 7 | \( 1 + (-0.818 - 0.573i)T \) |
| 11 | \( 1 + (-0.954 + 0.298i)T \) |
| 13 | \( 1 + (0.486 - 0.873i)T \) |
| 17 | \( 1 + (0.690 - 0.722i)T \) |
| 19 | \( 1 + (-0.518 - 0.855i)T \) |
| 23 | \( 1 + (-0.275 + 0.961i)T \) |
| 29 | \( 1 + (0.783 - 0.620i)T \) |
| 31 | \( 1 + (0.763 - 0.645i)T \) |
| 37 | \( 1 + (0.593 - 0.804i)T \) |
| 41 | \( 1 + (-0.366 + 0.930i)T \) |
| 43 | \( 1 + (-0.545 + 0.838i)T \) |
| 47 | \( 1 + (-0.992 + 0.121i)T \) |
| 53 | \( 1 + (0.540 + 0.841i)T \) |
| 59 | \( 1 + (-0.869 + 0.493i)T \) |
| 61 | \( 1 + (-0.994 + 0.100i)T \) |
| 67 | \( 1 + (-0.999 + 0.00531i)T \) |
| 71 | \( 1 + (-0.375 + 0.926i)T \) |
| 73 | \( 1 + (-0.999 + 0.0265i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.983 - 0.179i)T \) |
| 89 | \( 1 + (-0.986 - 0.164i)T \) |
| 97 | \( 1 + (-0.818 - 0.573i)T \) |
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
−18.49211665012719054095633509189, −17.83260216208081358972343521336, −16.621504896800209876340743362446, −16.48251676289216120466629515887, −15.75646053763402872259431153031, −14.95043567921220037072932756565, −13.9845035812384453483847429594, −13.43171338179236368887206167342, −12.82482914548756986824762831785, −12.2256550336111392490285622533, −11.96956711398452698841575899712, −10.777038629673809238908060164190, −10.29148864153481116023578285810, −9.90394665883589128611628645387, −8.89062686480046734253496419536, −8.16870854902358804338892545615, −6.690633547222583368775025945151, −6.36113180211180463797416008646, −5.76619253442971244947257622278, −5.14548345765292805949321024147, −4.49270777649960397102181972974, −3.57371664548502103422578704620, −2.637917686066022529310765075997, −1.81365452885779279268462071590, −1.192322964296749034807816605260,
0.04809439442874939697062029514, 1.36953385145832218562451311648, 2.85426701203360979979276050182, 2.97660159142382340019610784758, 4.20501354396543470175159682134, 4.815312786680866143155799766047, 5.75808591479569699408190419722, 5.96897398761558799147321709694, 6.77044274178363011568904580867, 7.38759520828762848462034499344, 8.01684373735660140813855506308, 9.373954031086754205431865357, 9.950295465095490986914447559506, 10.585422543948619753297652868444, 11.2382661508377135463223946164, 12.02820870536117358897934005593, 12.87609407634434942093975597902, 13.39436515159357757163081864495, 13.63419742910266537093441574351, 14.85000446940260042874471112805, 15.37077916295095942923605337305, 15.983515201553572671365772080473, 16.56042635846362400877806482139, 17.2726519974368179777667214575, 17.88515077996640954909047691886