L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + 10-s + (−0.173 − 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)17-s + 19-s + (0.766 − 0.642i)20-s + (−0.766 − 0.642i)22-s + (−0.173 + 0.984i)23-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + 10-s + (−0.173 − 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)17-s + 19-s + (0.766 − 0.642i)20-s + (−0.766 − 0.642i)22-s + (−0.173 + 0.984i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.878475344 + 0.2695909818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.878475344 + 0.2695909818i\) |
\(L(1)\) |
\(\approx\) |
\(1.555086153 - 0.3730941737i\) |
\(L(1)\) |
\(\approx\) |
\(1.555086153 - 0.3730941737i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.173 - 0.984i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.939 + 0.342i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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
−22.1901048459806443053817494336, −21.20209851435841843850244164869, −20.42165063793840795282327039396, −19.954276187582581383009738551715, −18.32809901531678678205190190487, −17.90206588112483509476202899045, −16.80097128343159043807114330791, −16.32022350333696063570400052850, −15.46079627413909257425804070342, −14.70548361358540306670605838986, −13.701800812959304032739386187743, −13.012755299654691674772232243327, −12.47388089950874821480683105398, −11.709095806866583335773736179249, −10.12388340549855466911350754716, −9.571359320462943416586005278350, −8.5690118845082488926457031008, −7.54141924960713158704304358078, −6.743759696395277432461200310352, −5.64192293316540518388989702041, −5.2826832107625757996191053036, −4.17160779621293825876196935430, −2.97816539480836166594405000432, −2.23631482256186238694368049465, −0.50473639873957173677096771772,
1.06543337712761851347826153712, 2.08672831288153007551288677599, 3.28087297303390291522367149777, 3.59790062440740104821830764044, 5.07590813206300269162662215031, 5.98653190995471614640661136969, 6.50515316684995432718045264286, 7.55378561309140135516430992526, 9.236686794179522118148452809719, 9.69441562849129352311182526483, 10.61620347807485084776603685619, 11.26857947571718187385903877260, 12.25383831801571446845060670372, 13.21888355663264800051764236230, 13.7825231294690101139835903604, 14.34676736817910509776589536193, 15.36798413221160237691295812728, 16.28294503948784330211632892333, 17.0630799896883019696554114043, 18.34315384620410717236608072779, 18.93184186050937806080559309604, 19.54018578780197214214133231243, 20.49469047706648970752142129604, 21.461884476770746491077572405485, 21.87992769082968266192155876746