| L(s) = 1 | − 4·3-s − 4·7-s + 8·9-s − 16-s + 16·21-s − 8·25-s − 12·27-s − 8·31-s + 24·37-s + 24·43-s + 4·48-s + 8·49-s − 24·61-s − 32·63-s − 16·67-s − 20·73-s + 32·75-s + 23·81-s + 32·93-s + 12·97-s + 4·103-s − 96·111-s + 4·112-s + 40·121-s + 127-s − 96·129-s + 131-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.51·7-s + 8/3·9-s − 1/4·16-s + 3.49·21-s − 8/5·25-s − 2.30·27-s − 1.43·31-s + 3.94·37-s + 3.65·43-s + 0.577·48-s + 8/7·49-s − 3.07·61-s − 4.03·63-s − 1.95·67-s − 2.34·73-s + 3.69·75-s + 23/9·81-s + 3.31·93-s + 1.21·97-s + 0.394·103-s − 9.11·111-s + 0.377·112-s + 3.63·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1351980439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1351980439\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−13.08387053589197631022679088028, −12.50343930246193362992094011565, −12.23668530395967147757095350018, −11.82289643901955104567121587191, −11.80062970170223287271475144603, −11.23367415944945866485573438577, −10.90862461343976034762190192532, −10.76674121723936565168200091958, −10.45471285693397489650266968309, −9.863472261309497591897124211760, −9.526070664807416336095550719717, −9.281829984004021987354363144779, −9.142985922861602836403785620908, −8.279743490344514498393476746399, −7.50373221530796780404781154817, −7.49688463317062462469160487759, −7.18222877076189872125708170246, −6.16709329434069627483400906865, −6.10595906160915434301636909297, −5.90669421672407264250503282656, −5.74767439564106201635358380998, −4.56412421266750816989800627129, −4.50368004225802633236355559239, −3.72172360716998279227991014408, −2.68850278398652390264720053394,
2.68850278398652390264720053394, 3.72172360716998279227991014408, 4.50368004225802633236355559239, 4.56412421266750816989800627129, 5.74767439564106201635358380998, 5.90669421672407264250503282656, 6.10595906160915434301636909297, 6.16709329434069627483400906865, 7.18222877076189872125708170246, 7.49688463317062462469160487759, 7.50373221530796780404781154817, 8.279743490344514498393476746399, 9.142985922861602836403785620908, 9.281829984004021987354363144779, 9.526070664807416336095550719717, 9.863472261309497591897124211760, 10.45471285693397489650266968309, 10.76674121723936565168200091958, 10.90862461343976034762190192532, 11.23367415944945866485573438577, 11.80062970170223287271475144603, 11.82289643901955104567121587191, 12.23668530395967147757095350018, 12.50343930246193362992094011565, 13.08387053589197631022679088028
