| L(s) = 1 | − 12·11-s − 8·25-s − 8·37-s − 16·49-s − 48·83-s − 32·97-s − 24·107-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.61·11-s − 8/5·25-s − 1.31·37-s − 2.28·49-s − 5.26·83-s − 3.24·97-s − 2.32·107-s + 7.81·121-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 + 1.23·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2820539744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2820539744\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.77903101710874446582898571255, −6.68259322183065217844392105458, −6.28507150974543660232269057913, −5.73354480490664087015383566364, −5.73230646352994958613664358022, −5.70324226126238849653780185111, −5.45598037436489630040522511266, −5.37073511413163294598744345430, −4.86906495872609416435022107462, −4.75954510626149368388795380823, −4.73901129700221629238549793581, −4.16453011900442013822460644010, −4.13030854861657757893314458726, −3.81343859589674030550127513939, −3.54664289994453280483871820388, −3.04222734005125359646225355926, −2.90373584091771913066129078834, −2.86482435180074017453468120525, −2.59159959974899513571196271239, −2.26280577542139297460551830189, −1.82742892418122828755979362369, −1.51207420712644413207972932992, −1.50399431092693880616321279907, −0.47832747922043475298689053588, −0.15912832029506996456657498644,
0.15912832029506996456657498644, 0.47832747922043475298689053588, 1.50399431092693880616321279907, 1.51207420712644413207972932992, 1.82742892418122828755979362369, 2.26280577542139297460551830189, 2.59159959974899513571196271239, 2.86482435180074017453468120525, 2.90373584091771913066129078834, 3.04222734005125359646225355926, 3.54664289994453280483871820388, 3.81343859589674030550127513939, 4.13030854861657757893314458726, 4.16453011900442013822460644010, 4.73901129700221629238549793581, 4.75954510626149368388795380823, 4.86906495872609416435022107462, 5.37073511413163294598744345430, 5.45598037436489630040522511266, 5.70324226126238849653780185111, 5.73230646352994958613664358022, 5.73354480490664087015383566364, 6.28507150974543660232269057913, 6.68259322183065217844392105458, 6.77903101710874446582898571255
