L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (−0.866 + 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.587 − 0.809i)22-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (−0.866 + 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.587 − 0.809i)22-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9797356834 - 0.3754746417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9797356834 - 0.3754746417i\) |
\(L(1)\) |
\(\approx\) |
\(0.9644375316 - 0.2738050804i\) |
\(L(1)\) |
\(\approx\) |
\(0.9644375316 - 0.2738050804i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.207 - 0.978i)T \) |
| 3 | \( 1 + (-0.406 + 0.913i)T \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.994 + 0.104i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.207 + 0.978i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.743 + 0.669i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.994 + 0.104i)T \) |
| 53 | \( 1 + (0.406 - 0.913i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.978 + 0.207i)T \) |
| 67 | \( 1 + (-0.994 - 0.104i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.743 + 0.669i)T \) |
| 79 | \( 1 + (0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + (-0.978 - 0.207i)T \) |
| 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
−27.62888723248432350184982599245, −26.31226248080069941424487355309, −25.23467110904049414189483749197, −24.86428322398254778141164955264, −23.67316083671286632385706196042, −22.96051416868312839330105394139, −22.32267627014309838709104315791, −20.87880637136145921763750186236, −19.54181697260883308687219056757, −18.34606343788864219580842879292, −17.84029393272852577300893006295, −16.7326415128704686840507667258, −15.936459521831631197985218455478, −14.49305086103660622253177728428, −13.84966614422364655579423878851, −12.60980367436480750207658098755, −11.992306431937068742657561619716, −10.348161915998625828259974531120, −8.87884323516486944446832341981, −7.84820537560086676480917529368, −6.89051128254044758069299057910, −5.98074755432580654932757996615, −4.89924841055472443892533544472, −3.33208613500128635950758953479, −1.25579582554016724101579035613,
1.153922591516583840028484871834, 3.17509459750197425546593742920, 3.928526042596736706191642950937, 5.23399533252627905281514799607, 6.1878312021091750126474680406, 8.35964321649641854043577440104, 9.39899587898599298996893135550, 10.22225656838285018031429562739, 11.46070905129206708164386830559, 11.79494646582711032075991741034, 13.446248020562053105632907738358, 14.264762549884324171948309141656, 15.46608625588693029364927186819, 16.58050705137990438250454521295, 17.60942011524698062134216143039, 18.71730737556844110984952903520, 19.74873063981090145799540530390, 20.79938059248118932146659227030, 21.46204900896723960224692903372, 22.31058195949054914078662519519, 23.15836932776986208637314951544, 24.08213390232688284064118614590, 25.7281750325613110959411481324, 26.72799824608092510498775290082, 27.55989639009324196148622589682