| L(s) = 1 | + 4-s + 2·29-s − 2·41-s − 49-s + 2·61-s − 64-s − 4·89-s + 4·101-s + 4·109-s + 2·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s + 2·29-s − 2·41-s − 49-s + 2·61-s − 64-s − 4·89-s + 4·101-s + 4·109-s + 2·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.798501921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798501921\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.42687884755821449236328197771, −6.41942850476901812278213890261, −5.97718707270445562694465093864, −5.97069707077278172145685403557, −5.61437120787050721073140334051, −5.53832768037043317014931884735, −5.34766938053447448727312078869, −4.82817639706075395559889072690, −4.74227812154269239500503017342, −4.71331755537479533627062239772, −4.55322230219640414296150518662, −4.20695096743417008473848386482, −3.79322940597855463350109435216, −3.71583700336393557688251818921, −3.39485618204379045521709257352, −3.22519730833797185130962250527, −3.03371683191849019474949865838, −2.68579482946489720756044380160, −2.54675585906979557940636393260, −2.18796229760240517194017384653, −1.97204088140060324704623947880, −1.80059168552924175355866682923, −1.26358612528574433633198875633, −1.18425011225338546727816471569, −0.56670579021120350992513561776,
0.56670579021120350992513561776, 1.18425011225338546727816471569, 1.26358612528574433633198875633, 1.80059168552924175355866682923, 1.97204088140060324704623947880, 2.18796229760240517194017384653, 2.54675585906979557940636393260, 2.68579482946489720756044380160, 3.03371683191849019474949865838, 3.22519730833797185130962250527, 3.39485618204379045521709257352, 3.71583700336393557688251818921, 3.79322940597855463350109435216, 4.20695096743417008473848386482, 4.55322230219640414296150518662, 4.71331755537479533627062239772, 4.74227812154269239500503017342, 4.82817639706075395559889072690, 5.34766938053447448727312078869, 5.53832768037043317014931884735, 5.61437120787050721073140334051, 5.97069707077278172145685403557, 5.97718707270445562694465093864, 6.41942850476901812278213890261, 6.42687884755821449236328197771
