| L(s) = 1 | − 3·5-s − 7-s + 3·11-s − 2·13-s − 3·17-s + 19-s − 3·23-s + 5·25-s + 6·29-s − 14·31-s + 3·35-s + 37-s − 6·41-s + 4·43-s − 18·47-s − 6·49-s − 3·53-s − 9·55-s + 18·59-s − 2·61-s + 6·65-s − 14·67-s + 73-s − 3·77-s − 26·79-s − 12·83-s + 9·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 0.904·11-s − 0.554·13-s − 0.727·17-s + 0.229·19-s − 0.625·23-s + 25-s + 1.11·29-s − 2.51·31-s + 0.507·35-s + 0.164·37-s − 0.937·41-s + 0.609·43-s − 2.62·47-s − 6/7·49-s − 0.412·53-s − 1.21·55-s + 2.34·59-s − 0.256·61-s + 0.744·65-s − 1.71·67-s + 0.117·73-s − 0.341·77-s − 2.92·79-s − 1.31·83-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.682435589394762343993097391725, −8.559667703884861700765028652580, −8.067070738954450795111832135701, −7.67982755718067335406529723644, −7.18031004327649279169844722785, −7.00408895697570206789308721471, −6.62655164966622257901573642173, −6.20510743827946891125923125282, −5.63746391515603009460602137897, −5.24127588809493621386342214859, −4.68029688432423288141268881833, −4.36567633114570737319857894131, −3.80662682966081054569339583484, −3.68910135041117545302581857980, −2.97132786522700562284428208741, −2.67083263474190673979202311800, −1.71461973223273899886713053790, −1.37022987096987733962398149336, 0, 0,
1.37022987096987733962398149336, 1.71461973223273899886713053790, 2.67083263474190673979202311800, 2.97132786522700562284428208741, 3.68910135041117545302581857980, 3.80662682966081054569339583484, 4.36567633114570737319857894131, 4.68029688432423288141268881833, 5.24127588809493621386342214859, 5.63746391515603009460602137897, 6.20510743827946891125923125282, 6.62655164966622257901573642173, 7.00408895697570206789308721471, 7.18031004327649279169844722785, 7.67982755718067335406529723644, 8.067070738954450795111832135701, 8.559667703884861700765028652580, 8.682435589394762343993097391725
