L(s) = 1 | + (0.885 − 0.464i)2-s + (0.120 + 0.992i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (0.120 − 0.992i)8-s + (−0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.354 − 0.935i)11-s + (0.885 + 0.464i)12-s + (0.568 − 0.822i)13-s + (0.568 + 0.822i)14-s + (−0.970 − 0.239i)15-s + (−0.354 − 0.935i)16-s + (0.568 − 0.822i)17-s + ⋯ |
L(s) = 1 | + (0.885 − 0.464i)2-s + (0.120 + 0.992i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (0.120 − 0.992i)8-s + (−0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.354 − 0.935i)11-s + (0.885 + 0.464i)12-s + (0.568 − 0.822i)13-s + (0.568 + 0.822i)14-s + (−0.970 − 0.239i)15-s + (−0.354 − 0.935i)16-s + (0.568 − 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.444398663 + 0.3372884727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444398663 + 0.3372884727i\) |
\(L(1)\) |
\(\approx\) |
\(1.516129310 + 0.1937475403i\) |
\(L(1)\) |
\(\approx\) |
\(1.516129310 + 0.1937475403i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (0.885 - 0.464i)T \) |
| 3 | \( 1 + (0.120 + 0.992i)T \) |
| 5 | \( 1 + (-0.354 + 0.935i)T \) |
| 7 | \( 1 + (0.120 + 0.992i)T \) |
| 11 | \( 1 + (-0.354 - 0.935i)T \) |
| 13 | \( 1 + (0.568 - 0.822i)T \) |
| 17 | \( 1 + (0.568 - 0.822i)T \) |
| 19 | \( 1 + (-0.748 + 0.663i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.970 - 0.239i)T \) |
| 31 | \( 1 + (0.885 - 0.464i)T \) |
| 37 | \( 1 + (-0.748 + 0.663i)T \) |
| 41 | \( 1 + (-0.354 + 0.935i)T \) |
| 43 | \( 1 + (-0.354 + 0.935i)T \) |
| 47 | \( 1 + (-0.748 - 0.663i)T \) |
| 53 | \( 1 + (0.120 - 0.992i)T \) |
| 59 | \( 1 + (0.568 + 0.822i)T \) |
| 61 | \( 1 + (-0.748 + 0.663i)T \) |
| 67 | \( 1 + (0.885 + 0.464i)T \) |
| 71 | \( 1 + (0.120 - 0.992i)T \) |
| 73 | \( 1 + (0.568 + 0.822i)T \) |
| 83 | \( 1 + (0.568 - 0.822i)T \) |
| 89 | \( 1 + (0.120 + 0.992i)T \) |
| 97 | \( 1 + (-0.748 + 0.663i)T \) |
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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
−30.974314135009070177081040194429, −30.355158838814652840865889258738, −29.192511660741367526025004224202, −28.12284707059795202667744218577, −26.29149039917001539589701019099, −25.48487658129374492339140061669, −24.27737721897493208806901452724, −23.559186966218491439982638726227, −23.040770010246387161091115050407, −21.10978913751476242283353994857, −20.34223330381299476662480417187, −19.24001375040785574164851423531, −17.39865161084635814614820760199, −16.82249566436177805509331027670, −15.35225671178249942907824716439, −14.05751416964848187365453029488, −13.068213422128560198578325241059, −12.37162842067562318863031654605, −11.0596664852716683513840574637, −8.7245492089306210619523868508, −7.600266901038835589881114288, −6.66882797796405377067145256426, −5.04535137775583192083814468222, −3.78190725048554460251150191097, −1.73486454520547861344368694780,
2.74313011299499568697965106567, 3.49718211521312403325656159375, 5.169118415177485236621129315605, 6.17482063839635000387675903527, 8.24419634717127714705239525990, 9.89949106246472966827637353262, 10.96882968680004616498087417560, 11.729297894168102012296507341849, 13.40990767043205680400057267411, 14.72886538401639194738722702363, 15.28272084882474959630071841818, 16.32711881948154317373118358704, 18.450819908174021513469299097272, 19.28693820218310258002995518314, 20.79224442027076361667761277597, 21.4308473258473635600634954379, 22.52703258802071864509857555113, 23.109301877516537801440928705570, 24.76254561864511906087696242877, 25.80336783484392374309166782199, 27.22834154251162789682972655934, 27.93588703472278199030458635688, 29.23512290393008067534111952500, 30.300438310636192518205359519975, 31.557948620789156469648676512190