L(s) = 1 | + (−0.919 + 0.391i)2-s + 3-s + (0.692 − 0.721i)4-s + (0.987 + 0.160i)5-s + (−0.919 + 0.391i)6-s + (0.799 − 0.600i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (−0.996 − 0.0804i)11-s + (0.692 − 0.721i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.987 + 0.160i)15-s + (−0.0402 − 0.999i)16-s + (0.428 − 0.903i)17-s + ⋯ |
L(s) = 1 | + (−0.919 + 0.391i)2-s + 3-s + (0.692 − 0.721i)4-s + (0.987 + 0.160i)5-s + (−0.919 + 0.391i)6-s + (0.799 − 0.600i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (−0.996 − 0.0804i)11-s + (0.692 − 0.721i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.987 + 0.160i)15-s + (−0.0402 − 0.999i)16-s + (0.428 − 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.626057821 - 0.1479221063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626057821 - 0.1479221063i\) |
\(L(1)\) |
\(\approx\) |
\(1.223836237 + 0.01985722233i\) |
\(L(1)\) |
\(\approx\) |
\(1.223836237 + 0.01985722233i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (-0.919 + 0.391i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (0.987 + 0.160i)T \) |
| 7 | \( 1 + (0.799 - 0.600i)T \) |
| 11 | \( 1 + (-0.996 - 0.0804i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.428 - 0.903i)T \) |
| 19 | \( 1 + (-0.0402 + 0.999i)T \) |
| 23 | \( 1 + (-0.200 - 0.979i)T \) |
| 29 | \( 1 + (-0.748 - 0.663i)T \) |
| 31 | \( 1 + (0.885 + 0.464i)T \) |
| 37 | \( 1 + (-0.200 + 0.979i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.278 + 0.960i)T \) |
| 47 | \( 1 + (0.948 - 0.316i)T \) |
| 53 | \( 1 + (-0.632 + 0.774i)T \) |
| 59 | \( 1 + (-0.0402 - 0.999i)T \) |
| 61 | \( 1 + (-0.919 + 0.391i)T \) |
| 67 | \( 1 + (0.278 + 0.960i)T \) |
| 71 | \( 1 + (-0.845 - 0.534i)T \) |
| 73 | \( 1 + (0.987 - 0.160i)T \) |
| 79 | \( 1 + (-0.970 - 0.239i)T \) |
| 83 | \( 1 + (-0.632 + 0.774i)T \) |
| 89 | \( 1 + (0.120 + 0.992i)T \) |
| 97 | \( 1 + (-0.996 + 0.0804i)T \) |
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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
−23.9826663854892762079723281965, −21.85269123383785066612471639345, −21.47172119593226067412774695975, −20.92871628863674101234919603459, −20.07073894717141466469780614021, −19.11770618709360601653660387878, −18.4597446548167336448861885822, −17.69809184791544947324785439615, −16.92938557483372300764726971063, −15.73832429728742336444713844903, −15.02196670648687345227973366671, −14.00233112107491148623713525346, −13.08973084501498600937741583360, −12.299923594355818966220421006919, −11.1029396466186093976239612201, −10.1768344894994617510721027399, −9.37140840481091560067824444404, −8.753635404057203773352604520794, −7.87482536263550195089147253931, −7.033766570801980501389116280626, −5.67459988621665648827464922147, −4.45121903154863381246032263106, −2.99421845054456480354354007423, −2.138493841718534342172991999253, −1.56296381113818664840116627142,
1.113186095868227106161626796898, 2.20742995561209738838157605354, 2.96874143553880612701275138738, 4.75340093604230255083445823009, 5.66726021345034075691909033467, 6.93626092408782938364701275764, 7.81788178359533590552338246799, 8.296591288303112147877136806690, 9.49815917222877121602776532508, 10.22646637010587438927675444104, 10.66905623457320627773860413131, 12.22872872010395123249494927965, 13.46840530634034487679328264567, 14.16259542793968751058686210578, 14.825847874464802156124580411213, 15.71726428222030877335296308094, 16.75925405284706006335472549788, 17.5609693034515967892744578831, 18.422349149764433460029343000427, 18.83341246099198895693976196755, 20.21643265953150299646289117331, 20.664515612040476354851745288290, 21.155542408059879436590741243617, 22.5792040558135871586082176899, 23.70532887310790662567297715661