| L(s) = 1 | + 2·9-s + 8·19-s − 28·49-s − 5·81-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 16·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 1.83·19-s − 4·49-s − 5/9·81-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 1.22·171-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-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
| \(L(1)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.41804476956842995249193085505, −6.29271407655236427610744979862, −5.87252255427914855921912953853, −5.81113352523631552509028784381, −5.49856286604428260017892730429, −5.28303009069981396384737105043, −5.07377034194210019090358183916, −4.81486791749538356176553442683, −4.78530692609946303248675764718, −4.61765614760641143620334831032, −4.25428065289298049354292789830, −3.94142737416113330815536991514, −3.73192359101095674866413692047, −3.51019217127761944155256999009, −3.40654482431188597024711804757, −3.14716312487960982634276622200, −2.76257739523020073045330005363, −2.54982160730103367601115448735, −2.47819439129769698588424574198, −1.88440510793638165811270605177, −1.64630922937879904292858501678, −1.41288938563407289484571452043, −1.20863639602178718507715692453, −0.798144473446670988925298395105, −0.17013275168272142208483974471,
0.17013275168272142208483974471, 0.798144473446670988925298395105, 1.20863639602178718507715692453, 1.41288938563407289484571452043, 1.64630922937879904292858501678, 1.88440510793638165811270605177, 2.47819439129769698588424574198, 2.54982160730103367601115448735, 2.76257739523020073045330005363, 3.14716312487960982634276622200, 3.40654482431188597024711804757, 3.51019217127761944155256999009, 3.73192359101095674866413692047, 3.94142737416113330815536991514, 4.25428065289298049354292789830, 4.61765614760641143620334831032, 4.78530692609946303248675764718, 4.81486791749538356176553442683, 5.07377034194210019090358183916, 5.28303009069981396384737105043, 5.49856286604428260017892730429, 5.81113352523631552509028784381, 5.87252255427914855921912953853, 6.29271407655236427610744979862, 6.41804476956842995249193085505
