L(s) = 1 | + (0.278 + 0.960i)2-s + 3-s + (−0.845 + 0.534i)4-s + (−0.200 − 0.979i)5-s + (0.278 + 0.960i)6-s + (0.692 + 0.721i)7-s + (−0.748 − 0.663i)8-s + 9-s + (0.885 − 0.464i)10-s + (−0.632 + 0.774i)11-s + (−0.845 + 0.534i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.200 − 0.979i)15-s + (0.428 − 0.903i)16-s + (0.987 − 0.160i)17-s + ⋯ |
L(s) = 1 | + (0.278 + 0.960i)2-s + 3-s + (−0.845 + 0.534i)4-s + (−0.200 − 0.979i)5-s + (0.278 + 0.960i)6-s + (0.692 + 0.721i)7-s + (−0.748 − 0.663i)8-s + 9-s + (0.885 − 0.464i)10-s + (−0.632 + 0.774i)11-s + (−0.845 + 0.534i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.200 − 0.979i)15-s + (0.428 − 0.903i)16-s + (0.987 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.646775372 + 1.336652340i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646775372 + 1.336652340i\) |
\(L(1)\) |
\(\approx\) |
\(1.408503822 + 0.7278412955i\) |
\(L(1)\) |
\(\approx\) |
\(1.408503822 + 0.7278412955i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (0.278 + 0.960i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.200 - 0.979i)T \) |
| 7 | \( 1 + (0.692 + 0.721i)T \) |
| 11 | \( 1 + (-0.632 + 0.774i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.987 - 0.160i)T \) |
| 19 | \( 1 + (0.428 + 0.903i)T \) |
| 23 | \( 1 + (0.799 + 0.600i)T \) |
| 29 | \( 1 + (0.120 + 0.992i)T \) |
| 31 | \( 1 + (0.568 + 0.822i)T \) |
| 37 | \( 1 + (0.799 - 0.600i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.0402 - 0.999i)T \) |
| 47 | \( 1 + (-0.919 - 0.391i)T \) |
| 53 | \( 1 + (0.948 + 0.316i)T \) |
| 59 | \( 1 + (0.428 - 0.903i)T \) |
| 61 | \( 1 + (0.278 + 0.960i)T \) |
| 67 | \( 1 + (-0.0402 - 0.999i)T \) |
| 71 | \( 1 + (-0.996 - 0.0804i)T \) |
| 73 | \( 1 + (-0.200 + 0.979i)T \) |
| 79 | \( 1 + (0.885 + 0.464i)T \) |
| 83 | \( 1 + (0.948 + 0.316i)T \) |
| 89 | \( 1 + (-0.970 + 0.239i)T \) |
| 97 | \( 1 + (-0.632 - 0.774i)T \) |
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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
−23.21141587400649855097758569251, −22.09027815737244136193846576934, −21.28614344790355086060277419068, −20.86056288700602122328408424113, −19.80016341742618517691302201041, −19.10550411291674351195270173622, −18.60710760827177498513749278857, −17.67417799431403447586688258466, −16.392151121253445333893411204555, −14.91752290810495765300596067956, −14.689769862469277679296627843476, −13.651795936284712365049130723971, −13.310600845282896391201580019176, −11.80530092517257100075822417227, −11.1534579168218215942116383837, −10.224868286198184903795250013669, −9.5738109485776554294660134261, −8.30315797621149231247759261840, −7.632280429886573981182377514725, −6.45462039060697712758751503837, −4.8935946120514235747595099239, −4.06805757816487038336424787644, −3.037477153672984010205321217124, −2.4447836221934270343373319986, −1.09748351037103238056382532483,
1.36823996186351384063698495158, 2.81305949196567063334489588475, 3.89284897923258939849395287602, 5.20070973856318734479103744051, 5.28619755177237805102182935012, 7.18956367825407529057526262593, 7.869532986957355102930578469614, 8.45226247285751638089126592225, 9.3299166339987299138251023601, 10.173040469978322679590912854, 12.167914911326329675140689176153, 12.50661418070442207077236403326, 13.44341970966045936553493186586, 14.435173048516827654271955785775, 15.098659285950187984850479878530, 15.72064281825011731696108647761, 16.59486987668541220738473860581, 17.69312741777267014773390172423, 18.33866084474977553262721877801, 19.3520579802127226278490297667, 20.49045376641714407266474979680, 20.994877230291211329704254085882, 21.773255941418910511502392964939, 23.0226101587691282131847695597, 23.75069954513027348740888207154