| L(s) = 1 | − 8·29-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 + 251-s − 256-s + 257-s + ⋯ |
| L(s) = 1 | − 8·29-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 + 251-s − 256-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2593600009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2593600009\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( 1 + T^{16} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{2} )^{8} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.67944899595411665374351538455, −3.65248546438315720795501947711, −3.53616052785617434231284650841, −3.50899077559317150139817199044, −3.42848620118638081251626539555, −3.36890813792578217101435048480, −3.31734818752044343136295401011, −3.03539749487068264259014779305, −2.89150343503455297486479455389, −2.64572421965100395777703633546, −2.63925417253509428192053180123, −2.52023263349646120232270109345, −2.30313131716569779227511852957, −2.19364978606981650138977199021, −2.09449614279751378486119373113, −1.96503259662278330604248039532, −1.84226536735857629298200980675, −1.82383472879881804538815198795, −1.62959880816502609777577457428, −1.35014428666364414839121969713, −1.31102616761306160534499402747, −1.09395995795140638738284172965, −1.00043546726594519778228972432, −0.50676112499730365368985937610, −0.16760731189300189861909321481,
0.16760731189300189861909321481, 0.50676112499730365368985937610, 1.00043546726594519778228972432, 1.09395995795140638738284172965, 1.31102616761306160534499402747, 1.35014428666364414839121969713, 1.62959880816502609777577457428, 1.82383472879881804538815198795, 1.84226536735857629298200980675, 1.96503259662278330604248039532, 2.09449614279751378486119373113, 2.19364978606981650138977199021, 2.30313131716569779227511852957, 2.52023263349646120232270109345, 2.63925417253509428192053180123, 2.64572421965100395777703633546, 2.89150343503455297486479455389, 3.03539749487068264259014779305, 3.31734818752044343136295401011, 3.36890813792578217101435048480, 3.42848620118638081251626539555, 3.50899077559317150139817199044, 3.53616052785617434231284650841, 3.65248546438315720795501947711, 3.67944899595411665374351538455
Plot not available for L-functions of degree greater than 10.
