L(s) = 1 | + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)7-s − 11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s − 31-s + (−0.173 − 0.984i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.173 − 0.984i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)7-s − 11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s − 31-s + (−0.173 − 0.984i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.173 − 0.984i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8195936879 - 0.3056641634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8195936879 - 0.3056641634i\) |
\(L(1)\) |
\(\approx\) |
\(0.7629066717 + 0.02295059559i\) |
\(L(1)\) |
\(\approx\) |
\(0.7629066717 + 0.02295059559i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.173 - 0.984i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.766 - 0.642i)T \) |
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
−22.67838774954493121523018611430, −21.87980211077776979287810841752, −20.87509006356302169171375809178, −19.890254614259228526279658633882, −19.71108274624281871195932978454, −18.30853789248183485729330507973, −17.9381309045398468521870179089, −16.689034475235860629281872476033, −16.12578554048794808974796041141, −14.90931050288874855389154031044, −14.67507370277737554800266694821, −13.36141818711538676646671341105, −12.65438433308569885114914109832, −11.62278383912865985308686662827, −10.72634381789559406112655362691, −10.323002843031343693420038455018, −8.89449789229145274210451973762, −7.85940796186507324596903509076, −7.43768321435120721455497061360, −6.41997936123625080399973077329, −4.959689229110844933066215031771, −4.3740012568914825362656478842, −3.240399765673451593521534616200, −2.20946702711478757972665189695, −0.62322893380308271629810156062,
0.3287051632575334646403450751, 1.97113855502436959055655488750, 2.85097777218757960831439897902, 4.210635720441874210471919225934, 4.96006691803348446525530778018, 5.8341775251052280203494463952, 7.28757791619958953746299398450, 7.83237798860463986329999668413, 8.808446183270720758993883259684, 9.56717558125881560462118878580, 10.88386290973131084861897133733, 11.57319111186189792164158138938, 12.314125981737682627808445615723, 13.11702988696550973971886438387, 14.22293370747002173933256164960, 15.32668944944952261831775068164, 15.54816641442877043899324987810, 16.573917661034820437762633809046, 17.59184126549980764872589252948, 18.377806059516930339276202119134, 19.185725991496688296426698030422, 19.93185797475400630829388848786, 20.79166985882150356089065924352, 21.60544182915979967388895327697, 22.36055413751386495360205786348