L(s) = 1 | + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯ |
L(s) = 1 | + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001639347303 - 0.9565729093i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001639347303 - 0.9565729093i\) |
\(L(1)\) |
\(\approx\) |
\(0.6594003007 - 0.8011081680i\) |
\(L(1)\) |
\(\approx\) |
\(0.6594003007 - 0.8011081680i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 + (-0.994 - 0.104i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.406 - 0.913i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.913 + 0.406i)T \) |
| 31 | \( 1 + (0.951 + 0.309i)T \) |
| 37 | \( 1 + (-0.406 - 0.913i)T \) |
| 41 | \( 1 + (-0.994 + 0.104i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.994 + 0.104i)T \) |
| 61 | \( 1 + (-0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.743 + 0.669i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.207 - 0.978i)T \) |
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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
−28.666538150707835898522357432041, −27.61962134651934963480575863035, −26.66744156899574726873081532024, −25.890247054874402964400889949364, −24.8775485033277425497585219642, −23.52063133097561324765635508261, −22.92927602231507150816574679051, −22.09262197535635264824192126980, −21.025270859888364724732045887501, −20.11448797035437888587282232054, −18.91271692799615134100003929808, −17.0316676500884057682449961842, −16.49266033131012452687485054200, −15.50472109171192480357857421620, −14.98027547477550085628713882436, −13.56786462948177421188784525083, −12.34080051191676469835143614855, −11.53241760764039747576458021838, −10.01563044007222079062962935065, −8.75096355816263282304724742630, −7.67022638264980998536314269258, −6.20044538667062027911956163646, −5.13500127036628115423263388543, −3.84832659012374216554679429112, −3.245984150303625626476983563372,
0.66800215537234075228704215257, 2.619730655802812866010280264990, 3.511504724463247280123536455441, 5.135712636180915082058977596875, 6.554004443653284207813190181983, 7.303387487932824385056187755055, 8.9814024266695302841302905110, 10.48260674255451353050549126174, 11.59014060083879927960608964044, 12.33888909176450540059280611848, 13.25238216969035050757580733645, 14.245636128173166845151893042194, 15.42340483899733875255491477644, 16.53203088142569088332477135955, 18.26055303330765579064885891318, 19.0429525743737294758640379863, 19.69684769005065841912675129540, 20.6038422727630048282924731005, 22.31593665219344537804360135692, 22.79741908945885435817714185359, 23.63851400634842345441678075407, 24.50768476635599588956156134031, 25.66140831613982571810471506927, 26.95160757563986310408810449314, 28.2628910047775026566126496362