L(s) = 1 | + (−0.412 + 0.910i)3-s + (0.999 − 0.0327i)5-s + (−0.659 − 0.751i)9-s + (0.162 + 0.986i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.227 − 0.973i)19-s + (0.0654 − 0.997i)23-s + (0.997 − 0.0654i)25-s + (0.956 − 0.290i)27-s + (0.634 − 0.773i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.683 + 0.729i)37-s + ⋯ |
L(s) = 1 | + (−0.412 + 0.910i)3-s + (0.999 − 0.0327i)5-s + (−0.659 − 0.751i)9-s + (0.162 + 0.986i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.227 − 0.973i)19-s + (0.0654 − 0.997i)23-s + (0.997 − 0.0654i)25-s + (0.956 − 0.290i)27-s + (0.634 − 0.773i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.683 + 0.729i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.498428328 + 0.7658605050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.498428328 + 0.7658605050i\) |
\(L(1)\) |
\(\approx\) |
\(1.088688940 + 0.3314078506i\) |
\(L(1)\) |
\(\approx\) |
\(1.088688940 + 0.3314078506i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.412 + 0.910i)T \) |
| 5 | \( 1 + (0.999 - 0.0327i)T \) |
| 11 | \( 1 + (0.162 + 0.986i)T \) |
| 13 | \( 1 + (-0.881 + 0.471i)T \) |
| 17 | \( 1 + (0.991 + 0.130i)T \) |
| 19 | \( 1 + (-0.227 - 0.973i)T \) |
| 23 | \( 1 + (0.0654 - 0.997i)T \) |
| 29 | \( 1 + (0.634 - 0.773i)T \) |
| 31 | \( 1 + (0.965 - 0.258i)T \) |
| 37 | \( 1 + (-0.683 + 0.729i)T \) |
| 41 | \( 1 + (0.555 - 0.831i)T \) |
| 43 | \( 1 + (0.995 + 0.0980i)T \) |
| 47 | \( 1 + (0.793 + 0.608i)T \) |
| 53 | \( 1 + (-0.986 + 0.162i)T \) |
| 59 | \( 1 + (-0.849 + 0.528i)T \) |
| 61 | \( 1 + (0.812 - 0.582i)T \) |
| 67 | \( 1 + (0.910 + 0.412i)T \) |
| 71 | \( 1 + (0.980 + 0.195i)T \) |
| 73 | \( 1 + (-0.946 - 0.321i)T \) |
| 79 | \( 1 + (0.130 + 0.991i)T \) |
| 83 | \( 1 + (-0.290 + 0.956i)T \) |
| 89 | \( 1 + (-0.896 - 0.442i)T \) |
| 97 | \( 1 + (-0.707 - 0.707i)T \) |
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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
−19.92967720413517537418672113432, −19.1569970928809799036220929249, −18.68194674323520656501226277603, −17.77296317654183451223014014074, −17.28845613934091845652511986002, −16.658474349665807359503806523541, −15.88017457319625217897359011265, −14.47524586792350415826546537938, −14.16678838045261348486255431579, −13.425671468075108817981495952141, −12.5480642054766751127368477748, −12.11934465407328300985545408905, −11.08462495538800794079104045591, −10.36494208044736840295978662518, −9.61213929456385387267544174957, −8.60326965455704532319437363114, −7.831409376876228565647528300788, −7.0523024628795939103724869159, −6.10266108970936912860520419632, −5.64179568695677009916681189321, −4.92852222004151700145662569440, −3.38255887733910820948591243205, −2.62959622182937747041099581246, −1.6219106840350150185199608153, −0.84348740480017862699178351238,
0.90017755993444069188636866559, 2.2479691489834298068308548507, 2.88076366241265607327382031894, 4.29903729133565686521847969088, 4.71735837503671388551188729984, 5.57877571366239035836138869632, 6.414570558639358338755141294635, 7.11276611707753668398208842590, 8.36374826704493653749710829898, 9.30249102023357634118724219, 9.79638523583285526213107294254, 10.33644643642065781556198441067, 11.19768011977220687622966982891, 12.243789614120532716212680549148, 12.589394097878239118755318972135, 13.95100266481851839343731152130, 14.33914310984213855946414861913, 15.20790036190492703857719884406, 15.83495291981925735479737535765, 17.01892765493620694868670178998, 17.12251826830304430442206335589, 17.80175556835105811740505919560, 18.83203976928176059375168642578, 19.65337088204906408631789254670, 20.761355584485083473689934286652