L(s) = 1 | + (−0.568 − 0.822i)2-s + (0.748 + 0.663i)3-s + (−0.354 + 0.935i)4-s + (−0.120 − 0.992i)5-s + (0.120 − 0.992i)6-s + (0.568 + 0.822i)7-s + (0.970 − 0.239i)8-s + (0.120 + 0.992i)9-s + (−0.748 + 0.663i)10-s + (0.885 − 0.464i)11-s + (−0.885 + 0.464i)12-s + (−0.354 − 0.935i)13-s + (0.354 − 0.935i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (−0.970 − 0.239i)17-s + ⋯ |
L(s) = 1 | + (−0.568 − 0.822i)2-s + (0.748 + 0.663i)3-s + (−0.354 + 0.935i)4-s + (−0.120 − 0.992i)5-s + (0.120 − 0.992i)6-s + (0.568 + 0.822i)7-s + (0.970 − 0.239i)8-s + (0.120 + 0.992i)9-s + (−0.748 + 0.663i)10-s + (0.885 − 0.464i)11-s + (−0.885 + 0.464i)12-s + (−0.354 − 0.935i)13-s + (0.354 − 0.935i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (−0.970 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8076448307 - 0.1855149870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8076448307 - 0.1855149870i\) |
\(L(1)\) |
\(\approx\) |
\(0.9266676864 - 0.1812204119i\) |
\(L(1)\) |
\(\approx\) |
\(0.9266676864 - 0.1812204119i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (-0.568 - 0.822i)T \) |
| 3 | \( 1 + (0.748 + 0.663i)T \) |
| 5 | \( 1 + (-0.120 - 0.992i)T \) |
| 7 | \( 1 + (0.568 + 0.822i)T \) |
| 11 | \( 1 + (0.885 - 0.464i)T \) |
| 13 | \( 1 + (-0.354 - 0.935i)T \) |
| 17 | \( 1 + (-0.970 - 0.239i)T \) |
| 19 | \( 1 + (0.354 + 0.935i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.885 + 0.464i)T \) |
| 31 | \( 1 + (-0.885 - 0.464i)T \) |
| 37 | \( 1 + (-0.748 - 0.663i)T \) |
| 41 | \( 1 + (-0.885 + 0.464i)T \) |
| 43 | \( 1 + (-0.748 + 0.663i)T \) |
| 47 | \( 1 + (0.120 - 0.992i)T \) |
| 59 | \( 1 + (0.120 - 0.992i)T \) |
| 61 | \( 1 + (0.970 - 0.239i)T \) |
| 67 | \( 1 + (0.354 - 0.935i)T \) |
| 71 | \( 1 + (0.748 - 0.663i)T \) |
| 73 | \( 1 + (0.970 + 0.239i)T \) |
| 79 | \( 1 + (-0.568 + 0.822i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.970 - 0.239i)T \) |
| 97 | \( 1 + (0.120 + 0.992i)T \) |
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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
−33.55540775718924911806819147308, −32.56137471789226203015064695366, −31.02186506299169983132123535300, −30.26754792970793713272591917800, −28.93644978762750604572521266641, −27.21377738906331541052655423860, −26.45066375225246840722465595874, −25.632442774213483890416149161886, −24.30094964201270115439136221026, −23.593203112814408402149761232173, −22.14669259552242130017459607946, −20.074911813572558815277848881386, −19.34919191762355789521407868412, −18.075466295155321623859627265949, −17.290545139597125869293049913055, −15.44560388459183683917961770888, −14.35139296038827636425656086329, −13.765226544539118223479330618525, −11.53024212981669568204062414814, −9.975975537473391308936256513819, −8.60964437938733297124803916599, −7.17878629146485088986794590735, −6.69832886060314155438188989959, −4.1860063211993476318779617113, −1.868663850388552420465481779843,
1.931210076710114181387727034839, 3.65704360884561282988051720650, 5.06140021942381380473110257170, 8.07877101203797065710926631490, 8.77545005410362853125693426099, 9.867492550909264924208078768731, 11.443111235641030983421005065803, 12.621395206105386633515083305976, 14.07380643463102561970725095840, 15.66881372311756670890156669039, 16.81854342448587552716669925783, 18.19598412457773782927188965991, 19.71130553237255527638612115556, 20.2996102326873753893655165309, 21.44594545037521651226983399738, 22.2807693630387493670183392134, 24.60682121876999673589223957279, 25.279628697158936527752420594970, 26.93512112836440146555655005709, 27.54958447253423001666866996882, 28.432374549685964144008515731614, 29.86977007816732762454777574638, 31.201917233819546781908269839635, 31.78554945719924885941508027477, 32.99944790963146838695510487919