L(s) = 1 | + (−0.970 + 0.239i)2-s + (−0.748 − 0.663i)3-s + (0.885 − 0.464i)4-s + (0.568 + 0.822i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (−0.748 + 0.663i)8-s + (0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + (0.885 − 0.464i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (0.568 − 0.822i)16-s + (0.885 − 0.464i)17-s + ⋯ |
L(s) = 1 | + (−0.970 + 0.239i)2-s + (−0.748 − 0.663i)3-s + (0.885 − 0.464i)4-s + (0.568 + 0.822i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (−0.748 + 0.663i)8-s + (0.120 + 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + (0.885 − 0.464i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (0.568 − 0.822i)16-s + (0.885 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5085425018 - 0.1753052999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5085425018 - 0.1753052999i\) |
\(L(1)\) |
\(\approx\) |
\(0.6052266510 - 0.09269378688i\) |
\(L(1)\) |
\(\approx\) |
\(0.6052266510 - 0.09269378688i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (-0.970 + 0.239i)T \) |
| 3 | \( 1 + (-0.748 - 0.663i)T \) |
| 5 | \( 1 + (0.568 + 0.822i)T \) |
| 7 | \( 1 + (-0.748 - 0.663i)T \) |
| 11 | \( 1 + (0.568 - 0.822i)T \) |
| 13 | \( 1 + (0.885 - 0.464i)T \) |
| 17 | \( 1 + (0.885 - 0.464i)T \) |
| 19 | \( 1 + (-0.354 - 0.935i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.120 - 0.992i)T \) |
| 31 | \( 1 + (-0.970 + 0.239i)T \) |
| 37 | \( 1 + (-0.354 - 0.935i)T \) |
| 41 | \( 1 + (0.568 + 0.822i)T \) |
| 43 | \( 1 + (0.568 + 0.822i)T \) |
| 47 | \( 1 + (-0.354 + 0.935i)T \) |
| 53 | \( 1 + (-0.748 + 0.663i)T \) |
| 59 | \( 1 + (0.885 + 0.464i)T \) |
| 61 | \( 1 + (-0.354 - 0.935i)T \) |
| 67 | \( 1 + (-0.970 - 0.239i)T \) |
| 71 | \( 1 + (-0.748 + 0.663i)T \) |
| 73 | \( 1 + (0.885 + 0.464i)T \) |
| 83 | \( 1 + (0.885 - 0.464i)T \) |
| 89 | \( 1 + (-0.748 - 0.663i)T \) |
| 97 | \( 1 + (-0.354 - 0.935i)T \) |
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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
−31.22933225729110014720019257625, −29.58531398910177808152339074243, −28.8331934342464461451974731163, −28.050436221598819533115636193340, −27.42092297269630660417255030398, −25.75206930985426951420787197151, −25.332592204313806927217172380980, −23.75741921554143313687850764878, −22.343760829846651406518156448943, −21.21944455361436354205297322962, −20.54867731684138783536450608103, −19.06291207083484259463162818784, −17.91993745585884623552289440317, −16.79508310932586102854958504767, −16.28059695179844830481031298681, −14.98999722982031773301975261525, −12.69481532966258394713597795141, −12.00579280568597825702971768465, −10.50028193430843738505224823974, −9.49943376295343005608311632273, −8.75363409411150233726233382474, −6.6595760841322623850766245666, −5.57983625219725862907960885599, −3.69603675810424868601351865513, −1.5334427964803433990184099762,
1.04423942968282711145887922015, 2.96596485578092600739786149357, 5.82217146185363144523620512185, 6.58432894249349101289545692016, 7.60359781690986182138875524587, 9.32672900597427416977493533230, 10.658823281814912486087625030799, 11.27664193169556932614436230516, 13.053043506473191813521728442208, 14.24233824630328632652721032507, 15.94253241313451308137918870427, 16.9171106324099628817439764861, 17.803168857495536430537063886091, 18.83862836342336518846847790715, 19.533238751923913976470172885659, 21.21695154926155592309913850085, 22.65375078668766803960499921199, 23.487363311598451349269006971630, 24.85784126736185355960852316715, 25.64915039991913065153482261242, 26.723483007137815456333475062337, 27.85001686980563806882513009540, 29.00379156733842163939754299411, 29.761357550138103111772888901674, 30.3078400011827043236824584104