L(s) = 1 | + (0.0747 − 0.997i)2-s + (0.955 − 0.294i)3-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (0.826 + 0.563i)11-s + (−0.988 + 0.149i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)2-s + (0.955 − 0.294i)3-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (0.826 + 0.563i)11-s + (−0.988 + 0.149i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6215422964 - 0.7461063056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6215422964 - 0.7461063056i\) |
\(L(1)\) |
\(\approx\) |
\(0.8941903223 - 0.6667823158i\) |
\(L(1)\) |
\(\approx\) |
\(0.8941903223 - 0.6667823158i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (0.0747 - 0.997i)T \) |
| 3 | \( 1 + (0.955 - 0.294i)T \) |
| 5 | \( 1 + (-0.733 - 0.680i)T \) |
| 11 | \( 1 + (0.826 + 0.563i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| 17 | \( 1 + (0.365 - 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 29 | \( 1 + (0.623 + 0.781i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (0.0747 - 0.997i)T \) |
| 53 | \( 1 + (-0.988 - 0.149i)T \) |
| 59 | \( 1 + (-0.733 + 0.680i)T \) |
| 61 | \( 1 + (-0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.0747 + 0.997i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 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
−34.302293760013111500644272882216, −32.727204822126194225193426305633, −32.14269512298520859352989863229, −30.91330230915581376303564273965, −30.135891608855586420349522805, −27.73927594715139781661274750414, −26.921648412207451261975657978997, −26.12501733536833379423283729062, −24.94594218920471627265547408482, −23.94197961321592667493942042088, −22.503394330100626386590599281109, −21.57533839287893341219052462235, −19.6301004022053841555019424292, −18.94160418323464753567156365082, −17.26812926857122366541325120641, −15.85172560609709720951199265470, −14.86587965245352911219539018652, −14.15558590051888667775178865296, −12.560435470748406590762797966470, −10.46373422870415167549428655519, −8.92000199084025167245464411045, −7.84508320030159829778493793470, −6.6222983166209683958116040400, −4.50129761572855002434912630862, −3.20609248834328382833663615757,
1.66120189701890085511173596314, 3.456239851768577080933030946163, 4.699081667584628521138836757287, 7.43168652526564684449745036150, 8.823908334926926098982783810664, 9.77507150179969122072786599473, 11.809509776643290623425627911016, 12.56590217723395437377606096137, 13.9677101210085745767616755592, 15.0700471256074107720615258495, 16.953517507083550132858900930792, 18.60393323226349988787207209608, 19.61828974924320153022841107854, 20.285469144156621709858776157711, 21.41395226716727632966091140546, 22.953698928346765186115568942366, 24.13744133472030256425483977421, 25.3952546797193109641880564360, 27.03193801327451585101686334753, 27.59009391677608301906063798394, 29.15764850041286239214069589606, 30.131922132806117281779595984275, 31.46567552674061170748760079027, 31.66983281947063958025538686705, 33.050419127018849291811551559948