| L(s) = 1 | + 8·5-s + 10·9-s + 36·25-s − 20·37-s + 80·45-s − 2·49-s + 36·53-s + 43·81-s + 36·89-s − 112·97-s + 16·121-s + 120·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s − 160·185-s + 191-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 10/3·9-s + 36/5·25-s − 3.28·37-s + 11.9·45-s − 2/7·49-s + 4.94·53-s + 43/9·81-s + 3.81·89-s − 11.3·97-s + 1.45·121-s + 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 11.7·185-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(50.13287206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(50.13287206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T )^{8} \) |
| 11 | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + T^{2} + 24 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T^{2} - 186 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 59 T^{2} + 1440 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 61 T^{2} + 1644 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 64 T^{2} + 1950 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 23 T^{2} + 420 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 + 5 T + 72 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 20 T^{2} + 1350 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 136 T^{2} + 8190 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 112 T^{2} + 6366 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 160 T^{2} + 12174 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 79 T^{2} + 8004 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 71 | \( ( 1 - 157 T^{2} + 15576 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 148 T^{2} + 11382 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 76 T^{2} + 11814 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 188 T^{2} + 17862 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 14 T + p T^{2} )^{8} \) |
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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
−3.53681144375040520369354672281, −3.47476576770139093025533511832, −3.35712183382092659142139007506, −3.22227422723914938157136636310, −3.11281328180179687930943150964, −2.75236162477076652597812349716, −2.61749420530594857926966740206, −2.59515523528924104277757137237, −2.57815887670650098889363525246, −2.45194704748854224441357671051, −2.41235088578703562779177833664, −2.30408363649918705630629022900, −1.90460937519437429548723363827, −1.87611395885461623591898405428, −1.71511233437883871699605145644, −1.71181262053409375773188049614, −1.53453198551238137825158300293, −1.49827724444586148916649251610, −1.47211476505596434376245208342, −1.15576168662604330240142940026, −1.02409216387650821607527866552, −0.955071541098561729915568421218, −0.59514578955397419416280270835, −0.51974041728429775552642275325, −0.25120615327085028879572847832,
0.25120615327085028879572847832, 0.51974041728429775552642275325, 0.59514578955397419416280270835, 0.955071541098561729915568421218, 1.02409216387650821607527866552, 1.15576168662604330240142940026, 1.47211476505596434376245208342, 1.49827724444586148916649251610, 1.53453198551238137825158300293, 1.71181262053409375773188049614, 1.71511233437883871699605145644, 1.87611395885461623591898405428, 1.90460937519437429548723363827, 2.30408363649918705630629022900, 2.41235088578703562779177833664, 2.45194704748854224441357671051, 2.57815887670650098889363525246, 2.59515523528924104277757137237, 2.61749420530594857926966740206, 2.75236162477076652597812349716, 3.11281328180179687930943150964, 3.22227422723914938157136636310, 3.35712183382092659142139007506, 3.47476576770139093025533511832, 3.53681144375040520369354672281
Plot not available for L-functions of degree greater than 10.
