| L(s) = 1 | − 3·5-s + 7-s + 5·11-s + 4·13-s − 2·17-s + 6·19-s + 2·23-s + 5·25-s + 2·29-s + 31-s − 3·35-s − 10·37-s + 8·41-s + 8·43-s − 8·47-s − 6·49-s − 5·53-s − 15·55-s + 13·59-s + 8·61-s − 12·65-s + 14·67-s + 24·71-s + 6·73-s + 5·77-s − 11·79-s + 14·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 1.50·11-s + 1.10·13-s − 0.485·17-s + 1.37·19-s + 0.417·23-s + 25-s + 0.371·29-s + 0.179·31-s − 0.507·35-s − 1.64·37-s + 1.24·41-s + 1.21·43-s − 1.16·47-s − 6/7·49-s − 0.686·53-s − 2.02·55-s + 1.69·59-s + 1.02·61-s − 1.48·65-s + 1.71·67-s + 2.84·71-s + 0.702·73-s + 0.569·77-s − 1.23·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.669526750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669526750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2} \) |
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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
−9.416153014287731163027472630135, −8.962421937375295856226508627249, −8.356733452935611321057150255680, −8.304647735984438699971080435374, −7.85332633071761396445230730626, −7.46948932265496636066758905606, −6.89307018268139484048795957921, −6.59274706908449388768408057311, −6.50073102568510262144719779044, −5.72781908044739375882499644054, −5.14887989781366985249746366960, −5.02952466837027985802932710474, −4.27170206799831564883800232733, −3.96620775787689703993076471303, −3.49820701845681104605162530537, −3.40770573119574259545494478338, −2.52740724282455881403258108323, −1.85415287071160131709695593505, −1.07399164998130814971228065367, −0.74280266540032460151494313452,
0.74280266540032460151494313452, 1.07399164998130814971228065367, 1.85415287071160131709695593505, 2.52740724282455881403258108323, 3.40770573119574259545494478338, 3.49820701845681104605162530537, 3.96620775787689703993076471303, 4.27170206799831564883800232733, 5.02952466837027985802932710474, 5.14887989781366985249746366960, 5.72781908044739375882499644054, 6.50073102568510262144719779044, 6.59274706908449388768408057311, 6.89307018268139484048795957921, 7.46948932265496636066758905606, 7.85332633071761396445230730626, 8.304647735984438699971080435374, 8.356733452935611321057150255680, 8.962421937375295856226508627249, 9.416153014287731163027472630135
