L(s) = 1 | + 2·13-s + 2·17-s + 8·37-s + 8·53-s − 2·73-s + 81-s + 10·89-s − 2·97-s + 4·101-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 2·13-s + 2·17-s + 8·37-s + 8·53-s − 2·73-s + 81-s + 10·89-s − 2·97-s + 4·101-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(7.297684749\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.297684749\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 37 | \( ( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 53 | \( ( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 89 | \( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.78078493873032308245740174740, −3.52244590849793936171477582073, −3.48444567854195544248087335568, −3.31183418884635302665518492425, −3.16506333841467406646810723115, −3.15585883393548520857470050963, −3.08297345838531656918322175391, −2.99578827622824067597478935302, −2.90822105014798316506708198786, −2.48985672056177143594054329864, −2.37278246452391450312378259737, −2.25216586337053787055338854721, −2.21647296787341595526501109389, −2.19462611321061363461237086159, −2.18089425945419952803428963485, −2.14485206147288123826099364059, −2.02921698656194227111275542751, −1.45272634367953630068183288294, −1.17462569582701276217779496759, −1.11519974014613452199786838663, −1.09943522073885087795436848309, −1.06279979876293503187293731204, −0.876628085897858628262699573574, −0.796156898929570856522469151921, −0.69952285132200704004475906738,
0.69952285132200704004475906738, 0.796156898929570856522469151921, 0.876628085897858628262699573574, 1.06279979876293503187293731204, 1.09943522073885087795436848309, 1.11519974014613452199786838663, 1.17462569582701276217779496759, 1.45272634367953630068183288294, 2.02921698656194227111275542751, 2.14485206147288123826099364059, 2.18089425945419952803428963485, 2.19462611321061363461237086159, 2.21647296787341595526501109389, 2.25216586337053787055338854721, 2.37278246452391450312378259737, 2.48985672056177143594054329864, 2.90822105014798316506708198786, 2.99578827622824067597478935302, 3.08297345838531656918322175391, 3.15585883393548520857470050963, 3.16506333841467406646810723115, 3.31183418884635302665518492425, 3.48444567854195544248087335568, 3.52244590849793936171477582073, 3.78078493873032308245740174740
Plot not available for L-functions of degree greater than 10.