L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.5 + 0.866i)4-s + (0.669 − 0.743i)5-s + (−0.104 + 0.994i)6-s + (0.669 − 0.743i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.978 − 0.207i)10-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)15-s + (−0.5 − 0.866i)16-s + (−0.978 + 0.207i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.5 + 0.866i)4-s + (0.669 − 0.743i)5-s + (−0.104 + 0.994i)6-s + (0.669 − 0.743i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.978 − 0.207i)10-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)15-s + (−0.5 − 0.866i)16-s + (−0.978 + 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3690185010 - 0.7060252289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3690185010 - 0.7060252289i\) |
\(L(1)\) |
\(\approx\) |
\(0.5948931192 - 0.5189184190i\) |
\(L(1)\) |
\(\approx\) |
\(0.5948931192 - 0.5189184190i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 151 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (0.669 - 0.743i)T \) |
| 11 | \( 1 + (0.913 + 0.406i)T \) |
| 13 | \( 1 + (0.913 - 0.406i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.669 + 0.743i)T \) |
| 47 | \( 1 + (0.913 - 0.406i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.978 + 0.207i)T \) |
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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
−28.25701867508403990855467038858, −27.27941666293850841721457249085, −26.57963619597239593991725620131, −25.537994772065057866563683354728, −24.54776313769412839072646976199, −23.6438129854716456331072741958, −22.29419775830724824656516747278, −22.02154458320154584299064260048, −20.65507846891718251042952478906, −18.98485273781941830836607408438, −18.022659820157483427121239320992, −17.57895496721830954692144011988, −16.37761998024439815423053817862, −15.4687846069722183304540269651, −14.547314027920057683179319331135, −13.57301009026481922365827398452, −11.55028328506404197867071347727, −10.94220977885187671709821885597, −9.553473674549582125427830158416, −8.91767193810407762488538599662, −7.18679699310179093397317425200, −6.064183069947048409188485503125, −5.4842148975756001296328313540, −3.947380937728778007621978723843, −1.639710026012035193996290222170,
1.066154196336515001377192419683, 1.873965014104157665525650462214, 4.05192557081429061437203666588, 5.1652305910587271449454547573, 6.7062581552468036929130833031, 7.991211683061812807334345555584, 9.11235332045108017015952252931, 10.4080135449243615152843201406, 11.248206568559308188052712405756, 12.2881148437445716755196554765, 13.22107899298374593719321539059, 14.020111850582577359364263178183, 16.24127540010145996467173917244, 17.11657501285431096210569578793, 17.74670920152286220776898404659, 18.51327277128282140468470576994, 20.13142875941149328539245068740, 20.42025659870746020091484372725, 21.86102927291501823331854634616, 22.5557847737298016860650862187, 23.838903149505910567937164276556, 24.74098840019080193516062206585, 25.81086419111003581977142281205, 27.16902488135564858677250295885, 27.97675619377650051942962556171