| L(s) = 1 | + 8·7-s + 8·17-s + 8·19-s + 8·23-s + 16·37-s − 8·41-s − 8·47-s + 32·49-s − 16·53-s − 24·59-s + 24·61-s + 16·67-s + 8·79-s − 81-s + 16·83-s + 16·97-s − 16·101-s − 24·103-s − 16·107-s + 8·113-s + 64·119-s + 20·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1.94·17-s + 1.83·19-s + 1.66·23-s + 2.63·37-s − 1.24·41-s − 1.16·47-s + 32/7·49-s − 2.19·53-s − 3.12·59-s + 3.07·61-s + 1.95·67-s + 0.900·79-s − 1/9·81-s + 1.75·83-s + 1.62·97-s − 1.59·101-s − 2.36·103-s − 1.54·107-s + 0.752·113-s + 5.86·119-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.26535433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.26535433\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 72 T^{3} + 98 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 1040 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 312 T^{3} + 2978 T^{4} + 312 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1104 T^{3} + 9266 T^{4} + 1104 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1328 T^{3} + 13522 T^{4} - 1328 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 10654 T^{4} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1584 T^{3} + 19346 T^{4} - 1584 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 23334 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1808 T^{3} + 25282 T^{4} - 1808 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.38514805244472773532127543450, −6.13343764565283431600595224461, −5.76958066747826872500900362893, −5.68700197461110327394514312740, −5.56370713126356036073270614598, −5.11704714322053966869780228556, −5.05154136265732323334380230684, −4.95104015974222097601905151712, −4.70110202971340616131698127529, −4.68261408145690496172338354299, −4.51127988595861461211314334685, −3.86204079527618133808777938835, −3.83374161651693401755505455427, −3.46030897152036134199343301654, −3.39551204200633055059236222846, −3.18789920828220897582611850870, −2.71568732936989867718299680224, −2.54818157244804698862281116061, −2.30824053983584234581886527535, −2.00134761677204641743224147201, −1.41031973975677508909517634548, −1.38608741281272281377939910950, −1.28575455397945886724924922681, −0.956600909390099060321446041911, −0.51264724350266367320865596322,
0.51264724350266367320865596322, 0.956600909390099060321446041911, 1.28575455397945886724924922681, 1.38608741281272281377939910950, 1.41031973975677508909517634548, 2.00134761677204641743224147201, 2.30824053983584234581886527535, 2.54818157244804698862281116061, 2.71568732936989867718299680224, 3.18789920828220897582611850870, 3.39551204200633055059236222846, 3.46030897152036134199343301654, 3.83374161651693401755505455427, 3.86204079527618133808777938835, 4.51127988595861461211314334685, 4.68261408145690496172338354299, 4.70110202971340616131698127529, 4.95104015974222097601905151712, 5.05154136265732323334380230684, 5.11704714322053966869780228556, 5.56370713126356036073270614598, 5.68700197461110327394514312740, 5.76958066747826872500900362893, 6.13343764565283431600595224461, 6.38514805244472773532127543450
