| L(s) = 1 | + 2·9-s − 28·19-s + 48·29-s + 4·31-s − 10·49-s − 24·59-s + 16·61-s + 44·79-s + 81-s − 24·89-s + 48·101-s − 4·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s − 56·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 6.42·19-s + 8.91·29-s + 0.718·31-s − 1.42·49-s − 3.12·59-s + 2.04·61-s + 4.95·79-s + 1/9·81-s − 2.54·89-s + 4.77·101-s − 0.383·109-s + 8/11·121-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 + 2.15·169-s − 4.28·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.366790723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.366790723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 14 T^{2} + 315 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 28 T^{2} + 255 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 142 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 110 T^{2} + 6409 T^{4} + 324830 T^{6} + 13948420 T^{8} + 324830 p^{2} T^{10} + 6409 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 134 T^{2} + 8115 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 34 T^{2} - 1653 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 448 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 230 T^{2} + 30769 T^{4} + 3025190 T^{6} + 232161940 T^{8} + 3025190 p^{2} T^{10} + 30769 p^{4} T^{12} + 230 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 206 T^{2} + 22969 T^{4} + 1814654 T^{6} + 124424404 T^{8} + 1814654 p^{2} T^{10} + 22969 p^{4} T^{12} + 206 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 224 T^{2} + 23730 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 12 T - 52 T^{2} + 216 T^{3} + 17679 T^{4} + 216 p T^{5} - 52 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 116 T^{2} + 3750 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.73099737404411196638279110975, −3.64454348579112142209402078507, −3.63187069452635646305452547692, −3.62271980323650479100387564274, −3.48259487610441857146031951067, −2.98076790308135089271614853727, −2.96322776930228144192367428412, −2.86043950724343263632848795746, −2.80980008860033721471333605847, −2.76324959252780837221577245797, −2.51423798632188176553402574902, −2.35495714475570489550628458882, −2.34851132685536304594667467712, −2.21643398855174535431902912868, −2.08790827476004179991758968553, −1.99237552250492068938053354448, −1.48598821670041611753944954311, −1.47508839883264783770891687683, −1.44828326328093578499724458956, −1.44445726164703261822669429636, −0.881768654248735594349130122691, −0.77795509260581961541259531352, −0.64081314050773020178865364162, −0.50183536063450682874778043849, −0.22682634731559279361046729131,
0.22682634731559279361046729131, 0.50183536063450682874778043849, 0.64081314050773020178865364162, 0.77795509260581961541259531352, 0.881768654248735594349130122691, 1.44445726164703261822669429636, 1.44828326328093578499724458956, 1.47508839883264783770891687683, 1.48598821670041611753944954311, 1.99237552250492068938053354448, 2.08790827476004179991758968553, 2.21643398855174535431902912868, 2.34851132685536304594667467712, 2.35495714475570489550628458882, 2.51423798632188176553402574902, 2.76324959252780837221577245797, 2.80980008860033721471333605847, 2.86043950724343263632848795746, 2.96322776930228144192367428412, 2.98076790308135089271614853727, 3.48259487610441857146031951067, 3.62271980323650479100387564274, 3.63187069452635646305452547692, 3.64454348579112142209402078507, 3.73099737404411196638279110975
Plot not available for L-functions of degree greater than 10.
