L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s − i·17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s − 31-s + 33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s − i·17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s − 31-s + 33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.405110285 + 0.8421906946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405110285 + 0.8421906946i\) |
\(L(1)\) |
\(\approx\) |
\(1.307488098 + 0.3833988023i\) |
\(L(1)\) |
\(\approx\) |
\(1.307488098 + 0.3833988023i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.382 + 0.923i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
| 29 | \( 1 + (0.923 + 0.382i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.382 + 0.923i)T \) |
| 41 | \( 1 + (-0.707 - 0.707i)T \) |
| 43 | \( 1 + (0.923 - 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.923 + 0.382i)T \) |
| 59 | \( 1 + (0.382 + 0.923i)T \) |
| 61 | \( 1 + (-0.923 - 0.382i)T \) |
| 67 | \( 1 + (0.923 + 0.382i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (0.382 - 0.923i)T \) |
| 89 | \( 1 + (-0.707 + 0.707i)T \) |
| 97 | \( 1 + T \) |
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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
−25.15121979019150812851990450830, −24.20820283503842120908485464069, −23.2852263784638790192647848174, −22.359025884950717067129819499169, −21.34805087438800797187431144050, −20.15306159206075010676375212221, −19.7515010874929556693778435438, −19.0180252386170409919878187333, −17.72074582698721516779972091302, −17.02393006953814676783419244146, −15.74719269653693932435312557657, −14.8369517062626667623384489989, −14.11662653032791246306274179718, −12.84106903606136781347873854138, −12.67048713329860308972401917865, −10.98871982320731574384961378444, −9.9350204786052885561580644558, −9.11122338504512121383408929882, −8.0475278014384324857316534183, −7.04085704246141390671609115094, −6.29484605372882401656448786984, −4.50465614210576701749787320724, −3.54068956722520816142097098850, −2.48791762377257227207825760244, −1.02725720815402650071384167414,
1.741683334614964726426153812617, 2.94128850662452654458552894798, 3.83898005744582268649950552167, 5.06253868842920260449302369096, 6.42216914449246975339937005480, 7.40843430993545705624079421211, 8.81217696028533518900506929693, 9.22013768803244464337363195541, 10.18118598126680996015122897525, 11.554996565263570023400557474616, 12.45414972336431763450797588805, 13.635895333043893069101355942, 14.35273952961020673449615328868, 15.25375134212572093895293839848, 16.196759978239763607219623885699, 16.88735715970243759866321323763, 18.48402916690847353741275030492, 19.11452392759554216292839748494, 19.82768900285788932447824361942, 20.86661572359776019502611706123, 21.76550692977121592988974572239, 22.31356993608543427365359777972, 23.575695033304393584112270403869, 24.78868349477786523344469090468, 25.24346595067859671898246617991