L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + (−0.809 + 0.587i)20-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + (−0.809 + 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5956334762 - 0.9216429538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5956334762 - 0.9216429538i\) |
\(L(1)\) |
\(\approx\) |
\(0.8676961069 - 0.7641745552i\) |
\(L(1)\) |
\(\approx\) |
\(0.8676961069 - 0.7641745552i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.104 + 0.994i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.669 + 0.743i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.913 + 0.406i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (-0.104 - 0.994i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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
−32.03973890832587879666578490375, −30.86687288234043900866547074907, −29.50633087787813592264414110859, −27.974643805291777989564889177501, −27.005888576444419094454476543482, −26.134314163992792849642907900017, −25.26393789369063848316239544004, −24.51290079164217201935190843174, −22.74958518257749446903451692194, −22.01920516348369056300930124870, −20.88210281599245899490332384987, −19.42249536671613540438134301046, −18.16043402447972433584158880230, −17.06511523540811457951726690824, −15.9726447285893869301126831207, −14.7473388718454776197515841696, −14.148151928713256942027800453695, −12.92604332417501172100032129097, −10.49924691379127745090794619333, −9.664570789924731707619360331147, −8.54836105599322430033443532328, −7.1698320186905603412186746685, −5.686059566119153231509162336737, −4.45567490399429252483318831660, −2.63246400368537343570482040109,
1.560734045844362448759627019972, 2.6072534207345332465775049164, 4.38359373003624553415255489328, 6.2036100536532354434564028196, 8.04645232483687150235638968971, 9.144053508888540312574143194757, 10.12509242221868070317779543925, 11.85046229577707308855801031195, 12.89192617804966489701676045352, 13.693402431179139536437921327578, 14.773578955526280501713256707846, 17.0806289538135755987823294932, 17.811775033332159279237994723610, 19.121954441351829675048392651079, 19.822789513509559260369358682575, 21.09108326493364476911662267100, 21.74429271401698563364630471746, 23.389745838365380086255614835091, 24.46075428979846108219350892390, 25.70919276358779005627704782380, 26.53143567744537204719028207510, 28.047763402004510485256093265725, 29.05877510276049425518268090140, 29.7894778614515192892214026089, 30.70664648384572731116160276862