L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + 47-s + (0.5 + 0.866i)53-s − 55-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + 47-s + (0.5 + 0.866i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9307234134 + 0.7319977611i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9307234134 + 0.7319977611i\) |
\(L(1)\) |
\(\approx\) |
\(1.028951323 + 0.3273427634i\) |
\(L(1)\) |
\(\approx\) |
\(1.028951323 + 0.3273427634i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 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 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.59760104537785505627538062929, −24.89171477762671425549073654166, −24.13369363185951020920645466601, −23.12096196795302518864414317192, −22.0529899905132677572609084870, −21.11742582904237845387040353856, −20.42866734648046439579311351664, −19.42395614919157392789797884267, −18.28059313247104745860592418857, −17.44739238831012860484729548006, −16.39740321920841420136676851173, −15.770257850427291422785566547489, −14.33350215469590104809536752769, −13.51829422068910302560704310275, −12.58586754791832105847465186718, −11.643820580705836612716414701933, −10.29625929567332295928616007183, −9.49391400856429514963753798535, −8.336642731265479246262764487073, −7.46536868157096858431793236224, −5.724478771508052956180639485051, −5.318601258050000278865786828728, −3.75578160107996720261186832632, −2.412805860991776896265927031375, −0.85098915330414160606247879601,
1.84132306452370481496888135302, 2.86074120806129196251579720302, 4.300796265710727327236401714256, 5.53842608403778341080908612027, 6.74984303900189984035333506065, 7.48059036239243131417451370837, 8.953497700897823036788911422782, 10.014140287882965080255544114287, 10.73410142669487925368068575230, 11.966456818460327972522050653193, 12.96141329414556695286266977866, 14.15079212672271173268007251523, 14.76696270973422192475723564567, 15.84399384743612500466103837719, 17.041053916201799700950356434735, 17.900963417614093362445644395622, 18.697539915552691333422909292576, 19.67794155954096706743127774298, 20.76814121202765894909721328065, 21.763833297919308615395704676941, 22.373581535639595205165893246734, 23.489146007133657581309039822210, 24.2780108242615751622174517053, 25.62159719841934558495116032828, 26.030324269358581804362786261703