L(s) = 1 | + (−0.0825 + 0.996i)2-s + (−0.846 − 0.533i)3-s + (−0.986 − 0.164i)4-s + (0.0495 − 0.998i)5-s + (0.601 − 0.799i)6-s + (0.490 + 0.871i)7-s + (0.245 − 0.969i)8-s + (0.431 + 0.901i)9-s + (0.991 + 0.131i)10-s + (0.991 − 0.131i)11-s + (0.746 + 0.665i)12-s + (−0.999 + 0.0330i)13-s + (−0.909 + 0.416i)14-s + (−0.574 + 0.818i)15-s + (0.945 + 0.324i)16-s + (−0.956 + 0.293i)17-s + ⋯ |
L(s) = 1 | + (−0.0825 + 0.996i)2-s + (−0.846 − 0.533i)3-s + (−0.986 − 0.164i)4-s + (0.0495 − 0.998i)5-s + (0.601 − 0.799i)6-s + (0.490 + 0.871i)7-s + (0.245 − 0.969i)8-s + (0.431 + 0.901i)9-s + (0.991 + 0.131i)10-s + (0.991 − 0.131i)11-s + (0.746 + 0.665i)12-s + (−0.999 + 0.0330i)13-s + (−0.909 + 0.416i)14-s + (−0.574 + 0.818i)15-s + (0.945 + 0.324i)16-s + (−0.956 + 0.293i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7981424119 + 0.3635415568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7981424119 + 0.3635415568i\) |
\(L(1)\) |
\(\approx\) |
\(0.7406570574 + 0.2126189309i\) |
\(L(1)\) |
\(\approx\) |
\(0.7406570574 + 0.2126189309i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (-0.0825 + 0.996i)T \) |
| 3 | \( 1 + (-0.846 - 0.533i)T \) |
| 5 | \( 1 + (0.0495 - 0.998i)T \) |
| 7 | \( 1 + (0.490 + 0.871i)T \) |
| 11 | \( 1 + (0.991 - 0.131i)T \) |
| 13 | \( 1 + (-0.999 + 0.0330i)T \) |
| 17 | \( 1 + (-0.956 + 0.293i)T \) |
| 19 | \( 1 + (-0.768 + 0.639i)T \) |
| 23 | \( 1 + (0.997 + 0.0660i)T \) |
| 29 | \( 1 + (0.789 - 0.614i)T \) |
| 31 | \( 1 + (0.945 + 0.324i)T \) |
| 37 | \( 1 + (-0.277 - 0.960i)T \) |
| 41 | \( 1 + (0.546 + 0.837i)T \) |
| 43 | \( 1 + (-0.724 - 0.689i)T \) |
| 47 | \( 1 + (0.945 + 0.324i)T \) |
| 53 | \( 1 + (0.652 + 0.757i)T \) |
| 59 | \( 1 + (-0.677 + 0.735i)T \) |
| 61 | \( 1 + (0.922 + 0.386i)T \) |
| 67 | \( 1 + (0.652 + 0.757i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.277 - 0.960i)T \) |
| 79 | \( 1 + (-0.148 + 0.988i)T \) |
| 83 | \( 1 + (0.601 - 0.799i)T \) |
| 89 | \( 1 + (0.601 - 0.799i)T \) |
| 97 | \( 1 + (0.701 + 0.712i)T \) |
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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
−22.90835052521920802245483125886, −22.11224233791289660117432574407, −21.71243511935553265809666489511, −20.71689904294461844802910115051, −19.81389498565178992168637087677, −19.10408553391628280035834303775, −17.97888002866482248986977539834, −17.32686423398624844490055799073, −16.971185626165703601043392414818, −15.36499341496165397791402592848, −14.59833558326036005849123925273, −13.80188372766653754603662543169, −12.68139361081426436548552590723, −11.60190793837285254461335078794, −11.183992448086066423994481626924, −10.369296800945115149738863966017, −9.74460471311120784510592934474, −8.677824567297616915066273321564, −7.16028228416328788287777689645, −6.531215456491436998152554334164, −4.93811391991630405529241097538, −4.383734071787682750586098815699, −3.377642950982001451538669792149, −2.17758053821218790300542647520, −0.75091667712454827016926958102,
0.90803429837138678547305606418, 2.08273169944953384977692840339, 4.31748365354564691995704007854, 4.87213849982882374046350198630, 5.82398937290234394195844349255, 6.50728494435218691835572918097, 7.58837449987017335166724161803, 8.55619118871092058274689757737, 9.14538663015968416200027035766, 10.354299372743882421329928589318, 11.72111685743728507224018508905, 12.34244253248609197538803107634, 13.09386826751306896895721869708, 14.10777095306639293237793891308, 15.09858995265057527622337782785, 15.90111753197455512864868686951, 16.98667126424897481403942908312, 17.200881725081597179098469808141, 17.997385934311673412084182840783, 19.11682793504267911777169179629, 19.60948997164883747525479233187, 21.28422214280605186188736468253, 21.844422067981685334661630536845, 22.74016588310717395487729145100, 23.55176934663026584182223856284