L(s) = 1 | + 2·5-s + 3·25-s + 2·29-s + 2·37-s − 2·53-s + 2·61-s − 2·73-s − 81-s − 2·89-s + 4·97-s − 2·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 2·5-s + 3·25-s + 2·29-s + 2·37-s − 2·53-s + 2·61-s − 2·73-s − 81-s − 2·89-s + 4·97-s − 2·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.313013518\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.313013518\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.981561352064478990295532479198, −8.913067598529495488702094163069, −8.496464401155093086501324871858, −7.995737472845140081692412820653, −7.67040064438679314514119852193, −7.14994014115242988470709014104, −6.61266789467305997149464159372, −6.47376079546832560673368628954, −6.11218130779767829218663894678, −5.78028760957398732186724775059, −5.20662776530580697136081023284, −5.06694596616062340701323294831, −4.37015151238388350983553211857, −4.29669721301503063957824826984, −3.32047058723932312643792262325, −2.96644912324728096945516213202, −2.45970070691212981498848123866, −2.23163055592869945829565136046, −1.31579764839845224557079622792, −1.14820562126073964932029727565,
1.14820562126073964932029727565, 1.31579764839845224557079622792, 2.23163055592869945829565136046, 2.45970070691212981498848123866, 2.96644912324728096945516213202, 3.32047058723932312643792262325, 4.29669721301503063957824826984, 4.37015151238388350983553211857, 5.06694596616062340701323294831, 5.20662776530580697136081023284, 5.78028760957398732186724775059, 6.11218130779767829218663894678, 6.47376079546832560673368628954, 6.61266789467305997149464159372, 7.14994014115242988470709014104, 7.67040064438679314514119852193, 7.995737472845140081692412820653, 8.496464401155093086501324871858, 8.913067598529495488702094163069, 8.981561352064478990295532479198