| L(s) = 1 | − 4·3-s + 8·9-s + 12·11-s + 12·19-s − 20·27-s − 48·33-s − 12·43-s + 12·49-s − 48·57-s − 4·59-s + 36·67-s + 50·81-s − 12·83-s − 32·97-s + 96·99-s − 20·107-s − 24·113-s + 72·121-s + 127-s + 48·129-s + 131-s + 137-s + 139-s − 48·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s + 3.61·11-s + 2.75·19-s − 3.84·27-s − 8.35·33-s − 1.82·43-s + 12/7·49-s − 6.35·57-s − 0.520·59-s + 4.39·67-s + 50/9·81-s − 1.31·83-s − 3.24·97-s + 9.64·99-s − 1.93·107-s − 2.25·113-s + 6.54·121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.95·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850893908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850893908\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )( 1 + 24 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 + 1234 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 1294 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 + 4786 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - 46 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.86064128453798018634404726128, −6.70560504915944584189624283534, −6.67707227333713701272575466308, −6.63376834381554036946358940904, −6.34588595196539007447578118127, −5.83792865444966772395299209985, −5.63661904033346718017121378947, −5.57088252483637738116436209740, −5.45636271510916020479813888467, −5.23290258186996399404548335291, −5.01694568048205713192358262416, −4.51724982003431956238242800376, −4.29291193644150157724098194859, −4.01462721425882279248544942683, −4.00658069465890824964854811837, −3.57336545675302032102460671821, −3.56690237118909899099067929649, −2.92830861523978133311758417872, −2.90369493158231005466479429586, −2.17830728129228609676492663249, −1.68817088915288885029569801554, −1.58561885084444240047906352936, −1.21384734293322764981441273725, −0.862744730920471668379930644098, −0.48420951722120543289628319774,
0.48420951722120543289628319774, 0.862744730920471668379930644098, 1.21384734293322764981441273725, 1.58561885084444240047906352936, 1.68817088915288885029569801554, 2.17830728129228609676492663249, 2.90369493158231005466479429586, 2.92830861523978133311758417872, 3.56690237118909899099067929649, 3.57336545675302032102460671821, 4.00658069465890824964854811837, 4.01462721425882279248544942683, 4.29291193644150157724098194859, 4.51724982003431956238242800376, 5.01694568048205713192358262416, 5.23290258186996399404548335291, 5.45636271510916020479813888467, 5.57088252483637738116436209740, 5.63661904033346718017121378947, 5.83792865444966772395299209985, 6.34588595196539007447578118127, 6.63376834381554036946358940904, 6.67707227333713701272575466308, 6.70560504915944584189624283534, 6.86064128453798018634404726128
