L(s) = 1 | + (0.694 − 0.719i)2-s + (−0.957 − 0.288i)3-s + (−0.0365 − 0.999i)4-s + (0.997 + 0.0729i)5-s + (−0.872 + 0.489i)6-s + (0.581 − 0.813i)7-s + (−0.744 − 0.667i)8-s + (0.833 + 0.551i)9-s + (0.744 − 0.667i)10-s + (0.934 + 0.357i)11-s + (−0.252 + 0.967i)12-s + (−0.694 + 0.719i)13-s + (−0.181 − 0.983i)14-s + (−0.934 − 0.357i)15-s + (−0.997 + 0.0729i)16-s + (−0.520 − 0.853i)17-s + ⋯ |
L(s) = 1 | + (0.694 − 0.719i)2-s + (−0.957 − 0.288i)3-s + (−0.0365 − 0.999i)4-s + (0.997 + 0.0729i)5-s + (−0.872 + 0.489i)6-s + (0.581 − 0.813i)7-s + (−0.744 − 0.667i)8-s + (0.833 + 0.551i)9-s + (0.744 − 0.667i)10-s + (0.934 + 0.357i)11-s + (−0.252 + 0.967i)12-s + (−0.694 + 0.719i)13-s + (−0.181 − 0.983i)14-s + (−0.934 − 0.357i)15-s + (−0.997 + 0.0729i)16-s + (−0.520 − 0.853i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8540743245 - 1.163058073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8540743245 - 1.163058073i\) |
\(L(1)\) |
\(\approx\) |
\(1.074238565 - 0.7759639016i\) |
\(L(1)\) |
\(\approx\) |
\(1.074238565 - 0.7759639016i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (0.694 - 0.719i)T \) |
| 3 | \( 1 + (-0.957 - 0.288i)T \) |
| 5 | \( 1 + (0.997 + 0.0729i)T \) |
| 7 | \( 1 + (0.581 - 0.813i)T \) |
| 11 | \( 1 + (0.934 + 0.357i)T \) |
| 13 | \( 1 + (-0.694 + 0.719i)T \) |
| 17 | \( 1 + (-0.520 - 0.853i)T \) |
| 19 | \( 1 + (0.181 - 0.983i)T \) |
| 23 | \( 1 + (-0.934 + 0.357i)T \) |
| 29 | \( 1 + (-0.872 - 0.489i)T \) |
| 31 | \( 1 + (0.957 - 0.288i)T \) |
| 37 | \( 1 + (-0.181 + 0.983i)T \) |
| 41 | \( 1 + (-0.581 + 0.813i)T \) |
| 43 | \( 1 + (-0.0365 + 0.999i)T \) |
| 47 | \( 1 + (0.391 - 0.920i)T \) |
| 53 | \( 1 + (0.457 + 0.889i)T \) |
| 59 | \( 1 + (0.322 + 0.946i)T \) |
| 61 | \( 1 + (-0.520 + 0.853i)T \) |
| 67 | \( 1 + (0.957 + 0.288i)T \) |
| 71 | \( 1 + (0.872 + 0.489i)T \) |
| 73 | \( 1 + (0.109 - 0.994i)T \) |
| 79 | \( 1 + (-0.391 - 0.920i)T \) |
| 83 | \( 1 + (0.989 - 0.145i)T \) |
| 89 | \( 1 + (0.639 + 0.768i)T \) |
| 97 | \( 1 + (-0.989 - 0.145i)T \) |
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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
−27.69588335121419913031094438665, −26.83123909354831349339945924051, −25.58994840485868174129156657624, −24.52423687365407675706490153653, −24.27223635109739072550945304083, −22.70742501421540470901179708185, −22.05920249216389526334593266316, −21.53552759170434421008225084492, −20.495301059394595091517754879802, −18.54125160273255520359941671176, −17.50120337475114156436862093333, −17.1225639002883875635479629303, −15.96144247182325625543362735304, −14.894993771810305280645861774099, −14.09018329289259623305130188063, −12.64310494356244487092951136212, −12.08312498775940046267169914937, −10.78404209101740260285149770279, −9.4547043921702638820133576391, −8.26723835303681227065857604374, −6.67421205405713481962073826652, −5.80019474226673481089339094419, −5.17878260026623028223426807257, −3.84118411635494647592514672859, −1.99310638899563247917122386012,
1.20956109695853529432662497208, 2.26558195496639146590604753245, 4.305161280115565000550929780206, 5.02247861726583872257081876975, 6.33692171829863130885483079812, 7.11399882584394798054074360171, 9.42077119637239372304053695631, 10.17560202568824084344406840871, 11.3771758336219808157351437586, 11.92247991060861092597864863266, 13.3936341403782862042297314393, 13.830259179936580459389277661482, 15.04828304450475008148203241077, 16.64938462535941894796830139285, 17.50807761685822315089794048943, 18.29708147816718370143801000206, 19.58801752298381789678307288183, 20.55971091466876300186854391889, 21.700099690737450628983409641078, 22.21583474567681972988528574421, 23.165934742321005244586797372755, 24.29396826669117440506271659310, 24.67750373946769186106937402689, 26.430621410769935346764388003855, 27.57678184129348296634688218726