L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + (0.669 − 0.743i)20-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (0.809 − 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + (0.669 − 0.743i)20-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (0.809 − 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04904963476 + 0.3113909811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04904963476 + 0.3113909811i\) |
\(L(1)\) |
\(\approx\) |
\(0.6354983028 + 0.2716844762i\) |
\(L(1)\) |
\(\approx\) |
\(0.6354983028 + 0.2716844762i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.978 + 0.207i)T \) |
| 7 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.913 - 0.406i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.913 - 0.406i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.978 + 0.207i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.669 + 0.743i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.669 + 0.743i)T \) |
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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
−20.24850694731466841218377660005, −19.72131084270781107561462678964, −19.18189013466186000022868835822, −18.57850192589735481441709971722, −17.532644292007273016308136794250, −16.69379018165209470605297065308, −15.49623083941758206713759487369, −15.30522163716275161776883212511, −14.21815950106345913414995871732, −13.127429549051398148728706696595, −12.728063020817698616967620561340, −12.01971616098665302985471861714, −11.12044726366121916088216560375, −10.59380735030522721481780876869, −9.431736092346434500958611143528, −8.89538647570694658210513013383, −8.03464283635069419576897322476, −6.80624234036327864259320125577, −5.960650227607196580086678656955, −4.882322084501088751074940403504, −4.09749156039074483686441217777, −3.32539002958978028871626367622, −2.514607658055634193970916663990, −1.30152584918366044945584511486, −0.10510783388120757491376613747,
0.55741708060788603663185440874, 2.59042422677542885159640094273, 3.53972118133638899265475673541, 4.20197173216737037894911634604, 5.03151664029501827326093596130, 6.2523303795121938521371048952, 6.87322094536002184339336830634, 7.460694835604670752783384894735, 8.49909626119820847535862482569, 9.05983289976103311171712883173, 10.21161760593009177110510493176, 11.01116228693736574793140189076, 12.26777973365376312694753347539, 12.59176013848602899160179692238, 13.637528638143882047455242875502, 14.27942103263476686913441481051, 15.25390365211328127436255599292, 15.73355123098207874719243570952, 16.45599137927967913707614736749, 17.05669023698952083653882236035, 18.08340234773761090254898347310, 18.85226504700326007691322227277, 19.495319656506380740510734818346, 20.35523054137544678477194913270, 21.31746815934085027462465667263