L(s) = 1 | + (0.978 + 0.207i)2-s + (0.866 − 0.5i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (−0.104 + 0.994i)10-s + (−0.994 − 0.104i)11-s + (0.994 − 0.104i)12-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.669 − 0.743i)18-s + (0.743 − 0.669i)19-s + ⋯ |
L(s) = 1 | + (0.978 + 0.207i)2-s + (0.866 − 0.5i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (−0.104 + 0.994i)10-s + (−0.994 − 0.104i)11-s + (0.994 − 0.104i)12-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.669 − 0.743i)18-s + (0.743 − 0.669i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.826163187 + 0.5969694630i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.826163187 + 0.5969694630i\) |
\(L(1)\) |
\(\approx\) |
\(2.261077108 + 0.3028644306i\) |
\(L(1)\) |
\(\approx\) |
\(2.261077108 + 0.3028644306i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 11 | \( 1 + (-0.994 - 0.104i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.994 + 0.104i)T \) |
| 19 | \( 1 + (0.743 - 0.669i)T \) |
| 23 | \( 1 + (-0.978 - 0.207i)T \) |
| 29 | \( 1 + (-0.587 - 0.809i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 43 | \( 1 + (-0.309 - 0.951i)T \) |
| 47 | \( 1 + (-0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.406 + 0.913i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.669 - 0.743i)T \) |
| 67 | \( 1 + (0.406 - 0.913i)T \) |
| 71 | \( 1 + (0.587 - 0.809i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.866 - 0.5i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.743 + 0.669i)T \) |
| 97 | \( 1 + (0.587 + 0.809i)T \) |
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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
−25.32328448955276217634684331740, −24.45722163179279047423606272109, −23.84479291110165364605481614489, −22.60891584756666320264504414768, −21.69012666553430625050971294490, −20.86686429190709790160731127838, −20.35043114616669870643625474803, −19.59407771343952624084824023215, −18.433499613093662894288559185724, −16.718066413014760459123756210432, −16.123179337867392115276242401, −15.17024051879946308783826410649, −14.32917714410183727686563505953, −13.38390640786870740496656038062, −12.66835474292779501717684937181, −11.701432429139469079536115063058, −10.13812527867191033716221998236, −9.74879070419712192301431596016, −8.13506135903944346917765864277, −7.4770621845879555045575493938, −5.56669367144506473463637131598, −4.99768115535640771799645937989, −3.87132037041735592119846515458, −2.799365034455296155077805284841, −1.64646403383423894731021058738,
2.05431779178180691512987905193, 2.83441808986625433956922436140, 3.74207462862302667739256855105, 5.22297013256674510899195200415, 6.39936719925363964080035264290, 7.42520748023360286694823117666, 7.883324010188719413242551073839, 9.60488301937690274760152869672, 10.64115115561381232745101726581, 11.88806419648972463617556684667, 12.71205942656196382755481968420, 13.85703711354859226151873585839, 14.25595832252517487716451810213, 15.201307151992873017083532136257, 15.96801296054379028578197819476, 17.42457789223705466368118767139, 18.46184567442406405568293226658, 19.33681736348669015128913272898, 20.24315825823051643597098597337, 21.245933248003707973154805712010, 21.91931334241858780189367836233, 23.00302553799654680276155016579, 23.825905629255279609502922620554, 24.57742487407348845047604935717, 25.561340221350548144005543517044