| L(s) = 1 | + 4·13-s − 2·19-s − 2·31-s − 2·37-s − 4·43-s + 2·67-s + 2·73-s − 2·79-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4·13-s − 2·19-s − 2·31-s − 2·37-s − 4·43-s + 2·67-s + 2·73-s − 2·79-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.035408875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035408875\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.35029311160322581954581122817, −6.05464373439526974414179292244, −5.84169771051646778767065579917, −5.74284353428467623929114088432, −5.56070801828993697873976343942, −5.21209064831300390378905128625, −5.05481438898857624395573915695, −4.93073349739182697894477832075, −4.71808483062269148402882622766, −4.26081058188177777243893952061, −4.22209425775612428703850884425, −3.89622480377167573362255900124, −3.67350046840134224942105899317, −3.64116853052654115677327253668, −3.38874110237557171302395999255, −3.26495586219787160874483694853, −3.17671985002054231632029758597, −2.54349786270367631372483893525, −2.16898883701288896554781900893, −2.15193637122187343174648184778, −1.89520071877361127533978749223, −1.38735657387229775615645375447, −1.32584255438721064839670523976, −1.26395969672897782674459178465, −0.35933448142704463429978873713,
0.35933448142704463429978873713, 1.26395969672897782674459178465, 1.32584255438721064839670523976, 1.38735657387229775615645375447, 1.89520071877361127533978749223, 2.15193637122187343174648184778, 2.16898883701288896554781900893, 2.54349786270367631372483893525, 3.17671985002054231632029758597, 3.26495586219787160874483694853, 3.38874110237557171302395999255, 3.64116853052654115677327253668, 3.67350046840134224942105899317, 3.89622480377167573362255900124, 4.22209425775612428703850884425, 4.26081058188177777243893952061, 4.71808483062269148402882622766, 4.93073349739182697894477832075, 5.05481438898857624395573915695, 5.21209064831300390378905128625, 5.56070801828993697873976343942, 5.74284353428467623929114088432, 5.84169771051646778767065579917, 6.05464373439526974414179292244, 6.35029311160322581954581122817
