L(s) = 1 | − 2-s + 3·5-s − 9-s − 3·10-s − 2·13-s − 2·17-s + 18-s + 6·25-s + 2·26-s − 2·29-s + 32-s + 2·34-s − 2·37-s − 2·41-s − 3·45-s − 49-s − 6·50-s + 3·53-s + 2·58-s − 61-s − 64-s − 6·65-s − 2·73-s + 2·74-s + 2·82-s − 6·85-s + 3·89-s + ⋯ |
L(s) = 1 | − 2-s + 3·5-s − 9-s − 3·10-s − 2·13-s − 2·17-s + 18-s + 6·25-s + 2·26-s − 2·29-s + 32-s + 2·34-s − 2·37-s − 2·41-s − 3·45-s − 49-s − 6·50-s + 3·53-s + 2·58-s − 61-s − 64-s − 6·65-s − 2·73-s + 2·74-s + 2·82-s − 6·85-s + 3·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2133322465\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2133322465\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−9.103478022571611103698751662218, −8.732058650871246485015262464303, −8.684704932872452075491222471335, −8.606203066781969818400924984915, −8.549347688283605609729142946833, −7.52202050583379250182937713569, −7.47111032825195387802704109835, −7.43672014559604176354689487191, −6.91410371957520061958330593574, −6.65723171380626883149697443706, −6.37443483358141840272337402600, −6.20536594115554305913447657607, −5.92043287394811455548647486446, −5.48251791481274009607292464551, −5.34800628034127901105470219065, −4.92728100258333235645881544918, −4.87925060063891447259781380224, −4.60106541389467582366599801433, −3.91980923375694544646322057930, −3.34431378684299440643763237606, −3.08670258509988933287752498379, −2.47511045532847256016510563952, −2.26072258942276964586639132378, −2.04615428070157945945136219029, −1.58874477707428449180824401754,
1.58874477707428449180824401754, 2.04615428070157945945136219029, 2.26072258942276964586639132378, 2.47511045532847256016510563952, 3.08670258509988933287752498379, 3.34431378684299440643763237606, 3.91980923375694544646322057930, 4.60106541389467582366599801433, 4.87925060063891447259781380224, 4.92728100258333235645881544918, 5.34800628034127901105470219065, 5.48251791481274009607292464551, 5.92043287394811455548647486446, 6.20536594115554305913447657607, 6.37443483358141840272337402600, 6.65723171380626883149697443706, 6.91410371957520061958330593574, 7.43672014559604176354689487191, 7.47111032825195387802704109835, 7.52202050583379250182937713569, 8.549347688283605609729142946833, 8.606203066781969818400924984915, 8.684704932872452075491222471335, 8.732058650871246485015262464303, 9.103478022571611103698751662218