L(s) = 1 | + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.623 − 0.781i)8-s + (−0.733 + 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.988 − 0.149i)16-s + (−0.900 − 0.433i)17-s − 19-s + (−0.0747 + 0.997i)22-s + (−0.0747 + 0.997i)23-s + (0.900 − 0.433i)26-s + (0.0747 + 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.826 + 0.563i)32-s + (−0.955 + 0.294i)34-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.623 − 0.781i)8-s + (−0.733 + 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.988 − 0.149i)16-s + (−0.900 − 0.433i)17-s − 19-s + (−0.0747 + 0.997i)22-s + (−0.0747 + 0.997i)23-s + (0.900 − 0.433i)26-s + (0.0747 + 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.826 + 0.563i)32-s + (−0.955 + 0.294i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.637658508 - 2.061078288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637658508 - 2.061078288i\) |
\(L(1)\) |
\(\approx\) |
\(1.277455013 - 0.6167794925i\) |
\(L(1)\) |
\(\approx\) |
\(1.277455013 - 0.6167794925i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.733 - 0.680i)T \) |
| 11 | \( 1 + (-0.733 + 0.680i)T \) |
| 13 | \( 1 + (0.955 + 0.294i)T \) |
| 17 | \( 1 + (-0.900 - 0.433i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T \) |
| 29 | \( 1 + (0.0747 + 0.997i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.900 + 0.433i)T \) |
| 41 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (0.988 + 0.149i)T \) |
| 47 | \( 1 + (-0.733 + 0.680i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 + (-0.365 - 0.930i)T \) |
| 61 | \( 1 + (-0.0747 - 0.997i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.955 - 0.294i)T \) |
| 89 | \( 1 + (0.222 - 0.974i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.83079942405666711705689287506, −18.99772035763537047579308030574, −18.047786901187490882358772149253, −17.63151774686130176506461564291, −16.57281796772083504595928373278, −16.14985721212137103995638556246, −15.35685308243871610539662766884, −14.82297360719168493010692350439, −13.864340811113025552937001678600, −13.27390990381192586666497093325, −12.7781643317053090003337497763, −11.87105603976159430454964458981, −10.92310805382346486633307765726, −10.51662100125155200632860727125, −9.05736450835282039966492841512, −8.43404417544709365292371486757, −7.91827835457463499754425522039, −6.83824924649214297748077303404, −6.12391348268996183334805346514, −5.6383625413386479950257426439, −4.45983662584382939475823143041, −4.02609622416472920128794455997, −2.89024833721031644429899749821, −2.26402144874680199957929425794, −0.70342102230576724499119496269,
0.45114462089240480227953820526, 1.5999234496787251459678179747, 2.330711100845661685178683498664, 3.188504587010742721235574586769, 4.20779631113957523258946382899, 4.68360288341061021711842898690, 5.698513181233122081855078006763, 6.38255835797587431166317687317, 7.22357338486036296435676346542, 8.26661978124519218407256689175, 9.23606049443344484975630262610, 9.828503303046611234348846104200, 10.88656086277160385140444659955, 11.13566031938273140428428993465, 12.08962312629687195640541737965, 13.002586153528921811381714596035, 13.25059754504923728059851762281, 14.17281051966508819134403677870, 14.92046605141111634347525451331, 15.66345199461060571317985425357, 16.11482119443209691746079809511, 17.39003070138251830439895505613, 18.0685620809563709482316821933, 18.74125441138815845680236754835, 19.46946437726323469412592299805