L(s) = 1 | + (0.980 − 0.197i)2-s + (0.717 − 0.696i)3-s + (0.922 − 0.386i)4-s + (−0.426 − 0.904i)5-s + (0.566 − 0.824i)6-s + (−0.311 − 0.950i)7-s + (0.827 − 0.561i)8-s + (0.0310 − 0.999i)9-s + (−0.596 − 0.802i)10-s + (0.00620 + 0.999i)11-s + (0.392 − 0.919i)12-s + (0.481 + 0.876i)13-s + (−0.492 − 0.870i)14-s + (−0.935 − 0.352i)15-s + (0.700 − 0.713i)16-s + (−0.215 − 0.976i)17-s + ⋯ |
L(s) = 1 | + (0.980 − 0.197i)2-s + (0.717 − 0.696i)3-s + (0.922 − 0.386i)4-s + (−0.426 − 0.904i)5-s + (0.566 − 0.824i)6-s + (−0.311 − 0.950i)7-s + (0.827 − 0.561i)8-s + (0.0310 − 0.999i)9-s + (−0.596 − 0.802i)10-s + (0.00620 + 0.999i)11-s + (0.392 − 0.919i)12-s + (0.481 + 0.876i)13-s + (−0.492 − 0.870i)14-s + (−0.935 − 0.352i)15-s + (0.700 − 0.713i)16-s + (−0.215 − 0.976i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.661520489 - 2.456560776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.661520489 - 2.456560776i\) |
\(L(1)\) |
\(\approx\) |
\(1.780601688 - 1.220442260i\) |
\(L(1)\) |
\(\approx\) |
\(1.780601688 - 1.220442260i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.980 - 0.197i)T \) |
| 3 | \( 1 + (0.717 - 0.696i)T \) |
| 5 | \( 1 + (-0.426 - 0.904i)T \) |
| 7 | \( 1 + (-0.311 - 0.950i)T \) |
| 11 | \( 1 + (0.00620 + 0.999i)T \) |
| 13 | \( 1 + (0.481 + 0.876i)T \) |
| 17 | \( 1 + (-0.215 - 0.976i)T \) |
| 19 | \( 1 + (0.0558 + 0.998i)T \) |
| 29 | \( 1 + (0.988 - 0.148i)T \) |
| 31 | \( 1 + (-0.0186 + 0.999i)T \) |
| 37 | \( 1 + (-0.873 - 0.487i)T \) |
| 41 | \( 1 + (0.105 - 0.994i)T \) |
| 43 | \( 1 + (-0.616 + 0.787i)T \) |
| 47 | \( 1 + (-0.917 - 0.398i)T \) |
| 53 | \( 1 + (-0.596 + 0.802i)T \) |
| 59 | \( 1 + (0.227 + 0.973i)T \) |
| 61 | \( 1 + (0.645 - 0.763i)T \) |
| 67 | \( 1 + (0.275 - 0.961i)T \) |
| 71 | \( 1 + (0.626 + 0.779i)T \) |
| 73 | \( 1 + (0.988 + 0.148i)T \) |
| 79 | \( 1 + (0.867 + 0.498i)T \) |
| 83 | \( 1 + (-0.834 + 0.551i)T \) |
| 89 | \( 1 + (0.503 - 0.863i)T \) |
| 97 | \( 1 + (-0.999 - 0.0372i)T \) |
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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
−23.70754547280112116746477533064, −22.60437720640198106004938805993, −22.0080794976012713614513434280, −21.57237143110216903950724518606, −20.59298924275762034248765139243, −19.5288942008343954518279694398, −19.127511093056549595183390727658, −17.84458117461854175155986149261, −16.497112912973948218623647801, −15.57560700287086332312477756675, −15.335555535206754700030721707534, −14.50706376470314955519669861895, −13.58961557975372377355924558453, −12.84391567256380418004393573222, −11.54013041445190230972372748849, −10.93825134377661202905780013172, −10.01506874452008781097952902649, −8.52302544568618241245388585949, −8.07172364863833354148056938992, −6.68493823148523747360194821199, −5.8782945562071817550545588354, −4.83368433883531151635296092781, −3.54782454827644520668627703694, −3.13796088408233038470989215474, −2.22428170236965383110727844363,
1.11536068959369810604886405436, 1.98622887156822905105758804225, 3.391157732092991760878336514870, 4.13057902218653279547063404505, 5.013357875593485867069784653105, 6.51978149605816169474467812500, 7.142249449491632956821003385661, 8.03004748832964351118326101587, 9.24509731506014907768934925043, 10.17172025517763497885941614647, 11.52613935853681137085949206334, 12.312430100745303026590742310761, 12.877152853144917244304730487747, 13.885331084363402622435229727650, 14.219588956974318281732893539104, 15.52559993134902558425265261260, 16.17097116253912900087240027679, 17.147128784818756972251022690218, 18.38257384896517266360287001650, 19.53384035823756350286162723737, 19.8991603860558442306361515816, 20.75391248664347687156603145260, 21.1324749273609068525112036844, 22.85124086351107572864949205594, 23.203838455040585234700859241538