| L(s) = 1 | + 8·5-s + 32·25-s + 8·29-s + 40·53-s − 8·97-s + 56·101-s − 20·121-s + 104·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 32/5·25-s + 1.48·29-s + 5.49·53-s − 0.812·97-s + 5.57·101-s − 1.81·121-s + 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.12198151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.12198151\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 2 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 478 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 9118 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−5.86332088725192208004827432347, −5.62088684908480081218184248278, −5.52484331164554630878857741091, −5.37296693200450229726460147129, −5.34011952202722946944095546029, −4.83046771803576638996628600374, −4.78185449926829167201559399371, −4.63961203850706955314039903208, −4.48700875227913410602053798968, −3.93939812765957951709140456319, −3.90584669789767551449935878160, −3.66949506639917000478347621673, −3.53101382779843561265039759705, −2.94268475704182839477958847624, −2.88745268463631758468322348208, −2.68700969302017390138803985455, −2.55717765230427145917686079670, −2.21522628638926707247004486682, −2.05838866376346706104409436290, −1.84258169864135345557101670043, −1.69774178315953902727445850644, −1.38292266183348848659444518008, −0.880369279923574797266791264317, −0.77839656183708008148516675506, −0.57710219870707966830547024501,
0.57710219870707966830547024501, 0.77839656183708008148516675506, 0.880369279923574797266791264317, 1.38292266183348848659444518008, 1.69774178315953902727445850644, 1.84258169864135345557101670043, 2.05838866376346706104409436290, 2.21522628638926707247004486682, 2.55717765230427145917686079670, 2.68700969302017390138803985455, 2.88745268463631758468322348208, 2.94268475704182839477958847624, 3.53101382779843561265039759705, 3.66949506639917000478347621673, 3.90584669789767551449935878160, 3.93939812765957951709140456319, 4.48700875227913410602053798968, 4.63961203850706955314039903208, 4.78185449926829167201559399371, 4.83046771803576638996628600374, 5.34011952202722946944095546029, 5.37296693200450229726460147129, 5.52484331164554630878857741091, 5.62088684908480081218184248278, 5.86332088725192208004827432347
