L(s) = 1 | + (−0.982 + 0.187i)2-s + (−0.998 + 0.0627i)3-s + (0.929 − 0.368i)4-s + (0.968 − 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.844 + 0.535i)8-s + (0.992 − 0.125i)9-s + (−0.904 + 0.425i)12-s + (−0.125 − 0.992i)13-s + (0.425 − 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.998 − 0.0627i)17-s + (−0.951 + 0.309i)18-s + (−0.968 + 0.248i)19-s + (0.535 − 0.844i)21-s + ⋯ |
L(s) = 1 | + (−0.982 + 0.187i)2-s + (−0.998 + 0.0627i)3-s + (0.929 − 0.368i)4-s + (0.968 − 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.844 + 0.535i)8-s + (0.992 − 0.125i)9-s + (−0.904 + 0.425i)12-s + (−0.125 − 0.992i)13-s + (0.425 − 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.998 − 0.0627i)17-s + (−0.951 + 0.309i)18-s + (−0.968 + 0.248i)19-s + (0.535 − 0.844i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1477093295 - 0.05752688639i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1477093295 - 0.05752688639i\) |
\(L(1)\) |
\(\approx\) |
\(0.3757473386 + 0.08494550299i\) |
\(L(1)\) |
\(\approx\) |
\(0.3757473386 + 0.08494550299i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.982 + 0.187i)T \) |
| 3 | \( 1 + (-0.998 + 0.0627i)T \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 13 | \( 1 + (-0.125 - 0.992i)T \) |
| 17 | \( 1 + (-0.998 - 0.0627i)T \) |
| 19 | \( 1 + (-0.968 + 0.248i)T \) |
| 23 | \( 1 + (-0.684 + 0.728i)T \) |
| 29 | \( 1 + (0.637 + 0.770i)T \) |
| 31 | \( 1 + (0.535 + 0.844i)T \) |
| 37 | \( 1 + (0.125 + 0.992i)T \) |
| 41 | \( 1 + (-0.992 + 0.125i)T \) |
| 43 | \( 1 + (-0.587 - 0.809i)T \) |
| 47 | \( 1 + (0.248 - 0.968i)T \) |
| 53 | \( 1 + (-0.248 + 0.968i)T \) |
| 59 | \( 1 + (-0.876 - 0.481i)T \) |
| 61 | \( 1 + (0.876 - 0.481i)T \) |
| 67 | \( 1 + (-0.998 - 0.0627i)T \) |
| 71 | \( 1 + (0.0627 + 0.998i)T \) |
| 73 | \( 1 + (0.982 - 0.187i)T \) |
| 79 | \( 1 + (-0.535 + 0.844i)T \) |
| 83 | \( 1 + (-0.844 + 0.535i)T \) |
| 89 | \( 1 + (0.425 - 0.904i)T \) |
| 97 | \( 1 + (0.368 + 0.929i)T \) |
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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
−20.71545120674118656652178060494, −19.69558100786318071189055432439, −19.219058777969761196318713275061, −18.4285190790212889035615513849, −17.57025403641662048502757779234, −17.09240080476340391705843463833, −16.37904928944520054081899031710, −15.877410361244496491619594270617, −14.91943861552427487174929188445, −13.622833150805235682075317757214, −12.87357227675495755841950852418, −12.0483433737230958883434893697, −11.31173380840947582884104478525, −10.64522147583086122525954574880, −9.98479076765388499072687112385, −9.244267624514435277114755301730, −8.21063342619442945566622131134, −7.257646206271152094347137015231, −6.47691494132025038661775616289, −6.20638401144331225280119186252, −4.54003428446350839892740550466, −3.99463501291224752856980054131, −2.523709726224301704832265986101, −1.622118741069803255320094511589, −0.42599546864655991508690787029,
0.11084742110900668757181417386, 1.351075974635305044115508555190, 2.36725147061461400224109125021, 3.44413848944987774831119903761, 4.89255359704918986894240751137, 5.65374046722776096526985620325, 6.44478357006246068426427666706, 6.954738069138870493384686394908, 8.16870645016341260740637144811, 8.81069128163861301538178370466, 9.87366422017925308532138025506, 10.33484616712710443894511053081, 11.1501038917744669261994234960, 12.03535647636182244008190758163, 12.51167986184592672202975525148, 13.53510411137891317259644960055, 14.99352003878490654653328338755, 15.53681077486401627879583078057, 16.00084089943371163341454546069, 16.988706570262876027536583944761, 17.4918809718051182176737558840, 18.27893330912276889983668293964, 18.75110870321249922185547527950, 19.71540480623976206680625252386, 20.30409105195138919139947066456