L(s) = 1 | + 8.72e5·31-s + 1.27e7·61-s + 2.67e6·81-s + 9.32e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | + 5.26·31-s + 7.19·61-s + 0.559·81-s + 4.78·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.1892358896\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1892358896\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.22478261057933853117071684318, −2.00865529148380358338303094750, −1.98256788533433913607024239800, −1.93246711820009649490783568032, −1.87075436122325522718108511206, −1.58940758707628016131669631518, −1.52299343315089617296054060062, −1.43630260126463667640317845530, −1.32222310641343908989677116673, −1.31122683927920804834651958202, −1.22184016080136453435783432396, −1.03983981346215821815666933255, −0.972905698602943728527516158442, −0.926755043266091466508941395935, −0.915939254120453771565684925970, −0.895852094849450602243651363148, −0.826455975817323715099989942604, −0.75215817795625197351574627996, −0.58960208945511585640683955098, −0.56910825273293113330278835741, −0.38681080182567198022772835507, −0.31572827895328921975065278808, −0.20326632402548862920240690671, −0.086364676113700296288276749004, −0.01547527579046980177481480587,
0.01547527579046980177481480587, 0.086364676113700296288276749004, 0.20326632402548862920240690671, 0.31572827895328921975065278808, 0.38681080182567198022772835507, 0.56910825273293113330278835741, 0.58960208945511585640683955098, 0.75215817795625197351574627996, 0.826455975817323715099989942604, 0.895852094849450602243651363148, 0.915939254120453771565684925970, 0.926755043266091466508941395935, 0.972905698602943728527516158442, 1.03983981346215821815666933255, 1.22184016080136453435783432396, 1.31122683927920804834651958202, 1.32222310641343908989677116673, 1.43630260126463667640317845530, 1.52299343315089617296054060062, 1.58940758707628016131669631518, 1.87075436122325522718108511206, 1.93246711820009649490783568032, 1.98256788533433913607024239800, 2.00865529148380358338303094750, 2.22478261057933853117071684318
Plot not available for L-functions of degree greater than 10.