L(s) = 1 | + (−0.872 + 0.489i)2-s + (−0.322 + 0.946i)3-s + (0.520 − 0.853i)4-s + (−0.457 + 0.889i)5-s + (−0.181 − 0.983i)6-s + (0.109 − 0.994i)7-s + (−0.0365 + 0.999i)8-s + (−0.791 − 0.611i)9-s + (−0.0365 − 0.999i)10-s + (−0.694 − 0.719i)11-s + (0.639 + 0.768i)12-s + (−0.872 + 0.489i)13-s + (0.391 + 0.920i)14-s + (−0.694 − 0.719i)15-s + (−0.457 − 0.889i)16-s + (−0.934 − 0.357i)17-s + ⋯ |
L(s) = 1 | + (−0.872 + 0.489i)2-s + (−0.322 + 0.946i)3-s + (0.520 − 0.853i)4-s + (−0.457 + 0.889i)5-s + (−0.181 − 0.983i)6-s + (0.109 − 0.994i)7-s + (−0.0365 + 0.999i)8-s + (−0.791 − 0.611i)9-s + (−0.0365 − 0.999i)10-s + (−0.694 − 0.719i)11-s + (0.639 + 0.768i)12-s + (−0.872 + 0.489i)13-s + (0.391 + 0.920i)14-s + (−0.694 − 0.719i)15-s + (−0.457 − 0.889i)16-s + (−0.934 − 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1066002789 - 0.08994092913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1066002789 - 0.08994092913i\) |
\(L(1)\) |
\(\approx\) |
\(0.3970325058 + 0.1447168962i\) |
\(L(1)\) |
\(\approx\) |
\(0.3970325058 + 0.1447168962i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.872 + 0.489i)T \) |
| 3 | \( 1 + (-0.322 + 0.946i)T \) |
| 5 | \( 1 + (-0.457 + 0.889i)T \) |
| 7 | \( 1 + (0.109 - 0.994i)T \) |
| 11 | \( 1 + (-0.694 - 0.719i)T \) |
| 13 | \( 1 + (-0.872 + 0.489i)T \) |
| 17 | \( 1 + (-0.934 - 0.357i)T \) |
| 19 | \( 1 + (0.391 - 0.920i)T \) |
| 23 | \( 1 + (-0.694 + 0.719i)T \) |
| 29 | \( 1 + (-0.181 + 0.983i)T \) |
| 31 | \( 1 + (-0.322 - 0.946i)T \) |
| 37 | \( 1 + (0.391 - 0.920i)T \) |
| 41 | \( 1 + (0.109 - 0.994i)T \) |
| 43 | \( 1 + (0.520 + 0.853i)T \) |
| 47 | \( 1 + (0.252 - 0.967i)T \) |
| 53 | \( 1 + (0.744 - 0.667i)T \) |
| 59 | \( 1 + (-0.976 - 0.217i)T \) |
| 61 | \( 1 + (-0.934 + 0.357i)T \) |
| 67 | \( 1 + (-0.322 + 0.946i)T \) |
| 71 | \( 1 + (-0.181 + 0.983i)T \) |
| 73 | \( 1 + (-0.997 + 0.0729i)T \) |
| 79 | \( 1 + (0.252 + 0.967i)T \) |
| 83 | \( 1 + (-0.581 + 0.813i)T \) |
| 89 | \( 1 + (0.833 - 0.551i)T \) |
| 97 | \( 1 + (-0.581 - 0.813i)T \) |
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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
−27.984851586666454194896086135794, −26.99359779694579940369439464023, −25.66309413596822497250868211957, −24.76260629641641885716875102254, −24.28118047691601193721862477757, −22.87562515038337253237998405266, −21.821440048131432958940379451246, −20.51569245051524936250791718329, −19.85446399456326714693574694462, −18.84531660307234152995673207027, −18.02494770637336410427372637609, −17.23672170094875575926139716234, −16.15999413982627625672386759022, −15.12647621632419633855785177209, −13.21396225896863690816929013992, −12.25749075826828533467238153365, −12.04385999168008796710060999511, −10.60712616237682764802703966109, −9.27201282877596475888988743070, −8.16830227117587700611843574903, −7.60579684468508244617239103002, −6.075703245460162086671571105813, −4.698274136759204049672659321451, −2.67194317892073381656898303733, −1.66218460649305053220576303635,
0.14821815346208261444795913912, 2.660028315176478317084167673862, 4.16039062986456380244577100111, 5.48117883778397761347747138499, 6.84092072252912059254285806333, 7.648587460473029141516534648943, 9.0671385211311893656863736814, 10.101686518765510625586511879659, 10.959494249482091708310409925134, 11.49915549488825553598036553806, 13.79284052203213456848593001898, 14.69472856999048810116996987089, 15.68718731155794926594664795027, 16.36366041542357148146614190232, 17.43364245258032151946428048443, 18.23414787840182449641828405778, 19.57141980322015907048906463691, 20.16378892881522272339722792066, 21.53058194587563251485989907993, 22.5217281486612611743616049897, 23.601911030372850106514350536341, 24.218665159633138125839439948433, 26.08691309324611677725106951040, 26.34637714449633202946744352025, 27.05352116292719488772517444043