L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·12-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)19-s + 21-s + (0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·12-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)19-s + 21-s + (0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.551259553 - 0.2330116587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.551259553 - 0.2330116587i\) |
\(L(1)\) |
\(\approx\) |
\(1.101414049 - 0.04009531447i\) |
\(L(1)\) |
\(\approx\) |
\(1.101414049 - 0.04009531447i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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
−31.64554960940056261916524014479, −30.76182724445534462827026964695, −29.87831324345929048348137097020, −28.431667159175101771898223614517, −27.350692876288771577421527960845, −26.708177162136176899345407041871, −25.56521567978437222102155486499, −24.6990278623724316081617306625, −22.90117261415345301163993207164, −21.11532339557252712770233749587, −20.82350653292517624681017295751, −19.67432242333197506117487775989, −18.503291720186764409647969964845, −17.31037127774154040306828698330, −16.04869609557190791825052716326, −14.80915012740775292218984120161, −13.45039575614290713323476016616, −11.880292365452453018864832367086, −10.45163854419782173049003507718, −9.63502682659947017905515903256, −8.164798769605277557717500803065, −7.411358846499888097325632622994, −4.67970656807767272745751884503, −3.14167488719527549263226040976, −1.58828228730185888672784363265,
1.18727734290306079392699140594, 2.78715456592901349322706702621, 5.32184199741746900133833028951, 6.928110549748028053019272037630, 8.20322687507862528928458476785, 8.8445954267281107897934137549, 10.4378334627757450355138171066, 11.90184348076496980626668076571, 13.67958549602665214299372169916, 14.74187654161974963414551601908, 15.715544503873203144701711214108, 17.26672539108407782781264824302, 18.43584447634860528441820994023, 19.090341485940538666135481514689, 20.392757758871029695627409087434, 21.40344014621555755854313195936, 23.59809435522986015224159779310, 24.355400631899760518864776517524, 25.240066625960190358797249319343, 26.32675332753800655881834558194, 27.1826321427485524478374934687, 28.42423871108982111935031872501, 29.61763604881879888337953950303, 30.75413019465097708150139912265, 31.935621178359423357546858338