L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.056391509 + 1.367615537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056391509 + 1.367615537i\) |
\(L(1)\) |
\(\approx\) |
\(1.202132386 + 0.8278040022i\) |
\(L(1)\) |
\(\approx\) |
\(1.202132386 + 0.8278040022i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−25.12466516619321023307065977013, −24.59757427417644218759631206246, −23.48145581500509637361077287348, −22.32146999213406539951792999559, −21.954509525617879442343786901314, −20.82211243096057944324051000121, −20.31695949753608962117404434787, −18.987774271164617918811715766164, −18.388344040572859649452715647962, −17.31024241156533136063893749064, −16.235782753712106003912853030018, −14.848476875758838922301974855691, −13.95385102338758835708172524732, −13.41808379690220550096183167327, −12.27990265758525294078408923671, −11.31862992836991316951301526563, −10.3428152815003180832846207498, −9.455337974816870880906600593022, −8.56678527179679518212046159687, −6.68275947806258230004709991475, −5.76847922261267732367280370332, −4.7909716816684875382086717122, −3.4301654841135520551571397405, −2.370925522613103881174261715333, −1.12100415854391069225694789742,
1.824816825704960166489948686006, 3.28762061245949480477879410764, 4.6236983765543234116531996381, 5.5287450340521253839243092658, 6.59670560024070417291167078274, 7.363904697776476789573771492631, 8.8518794643006606570948916236, 9.47263800853339197362253895657, 10.84624979770336633813797010875, 12.2677978891128792606202731742, 13.07131272170200320252722398921, 13.937927181308284960607132695238, 14.80458229769957771780827630225, 15.66126518261074046262737914727, 16.8579426276921012504008258877, 17.54994247149948111917823399732, 18.157402809036367270507551275687, 19.67388382202229462616689960684, 20.76687793028697030425460877902, 21.79379271853094569956748433532, 22.22674785654762854607795561296, 23.37349652108373616912254114739, 24.21875610059432672485698346565, 25.16331036623016307331560064627, 25.706813152168143760406640546918