| L(s) = 1 | + 2-s − 2·5-s − 2·10-s + 25-s − 32-s − 3·37-s + 41-s + 49-s + 50-s − 2·61-s − 64-s + 2·73-s − 3·74-s + 82-s − 5·97-s + 98-s − 5·101-s + 3·113-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s + 2·146-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2-s − 2·5-s − 2·10-s + 25-s − 32-s − 3·37-s + 41-s + 49-s + 50-s − 2·61-s − 64-s + 2·73-s − 3·74-s + 82-s − 5·97-s + 98-s − 5·101-s + 3·113-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s + 2·146-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5518670145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5518670145\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | | \( 1 \) |
| 41 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 5 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.92005070496333595305721909616, −6.85977102001624172645777669088, −6.55556378987915228958862767987, −6.50179232253351791409010201057, −6.06553867722742514705156852233, −5.59172616806811082318485630938, −5.58568469212997951086204220008, −5.56175773671396666437935041190, −5.24337484690584067268427806166, −4.85672581199951381243266400031, −4.75884150864223633637409600995, −4.45882581543749184009160549208, −4.33380897945912933840327352547, −3.89831503195044166278346366988, −3.81939552269756609021252289208, −3.81196484967507742801478338116, −3.59461404299543588763263681368, −3.06923791260437731753546415677, −2.88435145977956756112454133400, −2.71722464938743137587005246293, −2.31449071373157395602683197834, −1.72187692367802884207467527473, −1.67851517832390548962474327200, −1.22714986757612128596992060257, −0.39820041942904164398153222382,
0.39820041942904164398153222382, 1.22714986757612128596992060257, 1.67851517832390548962474327200, 1.72187692367802884207467527473, 2.31449071373157395602683197834, 2.71722464938743137587005246293, 2.88435145977956756112454133400, 3.06923791260437731753546415677, 3.59461404299543588763263681368, 3.81196484967507742801478338116, 3.81939552269756609021252289208, 3.89831503195044166278346366988, 4.33380897945912933840327352547, 4.45882581543749184009160549208, 4.75884150864223633637409600995, 4.85672581199951381243266400031, 5.24337484690584067268427806166, 5.56175773671396666437935041190, 5.58568469212997951086204220008, 5.59172616806811082318485630938, 6.06553867722742514705156852233, 6.50179232253351791409010201057, 6.55556378987915228958862767987, 6.85977102001624172645777669088, 6.92005070496333595305721909616
