| L(s) = 1 | + 6·3-s − 8·7-s + 18·9-s − 4·11-s − 6·13-s + 4·16-s + 6·17-s − 48·21-s + 8·23-s + 36·27-s − 24·33-s − 36·39-s − 36·41-s − 16·43-s − 10·47-s + 24·48-s + 32·49-s + 36·51-s + 6·53-s + 14·61-s − 144·63-s + 6·67-s + 48·69-s + 30·71-s + 4·73-s + 32·77-s + 45·81-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 3.02·7-s + 6·9-s − 1.20·11-s − 1.66·13-s + 16-s + 1.45·17-s − 10.4·21-s + 1.66·23-s + 6.92·27-s − 4.17·33-s − 5.76·39-s − 5.62·41-s − 2.43·43-s − 1.45·47-s + 3.46·48-s + 32/7·49-s + 5.04·51-s + 0.824·53-s + 1.79·61-s − 18.1·63-s + 0.733·67-s + 5.77·69-s + 3.56·71-s + 0.468·73-s + 3.64·77-s + 5·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.073611816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.073611816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} + 36 T^{3} + 23 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 - 8 T + 32 T^{2} + 112 T^{3} - 977 T^{4} + 112 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 55 T^{2} + 2184 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 16 T + 128 T^{2} + 672 T^{3} + 3527 T^{4} + 672 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 10 T + 50 T^{2} - 440 T^{3} - 4409 T^{4} - 440 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 2137 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 3649 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 552 T^{3} - 6433 T^{4} + 552 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 31 T^{2} - 6960 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 24 T + 288 T^{2} + 2304 T^{3} + 18431 T^{4} + 2304 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.124051740540613518143885596469, −7.82079323609303669575528495344, −7.76117048298562390801954091994, −7.06480837231137481220671639290, −7.02030675873192717083941645936, −6.83494534823652798190123119286, −6.73777162266986931519532680859, −6.64907161822454627203339368991, −6.08056953610029820667500758631, −5.56492780231055768386412787831, −5.32250508128364480329283485801, −5.20754950886094042229688863304, −4.89335176815721131915763373119, −4.74486587098875870812952531646, −3.87211134549189222819876143941, −3.57912673457605575047191677448, −3.53443123454865740533719985096, −3.47934768702709449299094979933, −2.97621162304248419906033745558, −2.94367650037530551376972800908, −2.77791834249907624658258737307, −2.31143175792073665535626074563, −1.91657498685519080018482092223, −1.49563199695168821457959949607, −0.43141828203135188143038543647,
0.43141828203135188143038543647, 1.49563199695168821457959949607, 1.91657498685519080018482092223, 2.31143175792073665535626074563, 2.77791834249907624658258737307, 2.94367650037530551376972800908, 2.97621162304248419906033745558, 3.47934768702709449299094979933, 3.53443123454865740533719985096, 3.57912673457605575047191677448, 3.87211134549189222819876143941, 4.74486587098875870812952531646, 4.89335176815721131915763373119, 5.20754950886094042229688863304, 5.32250508128364480329283485801, 5.56492780231055768386412787831, 6.08056953610029820667500758631, 6.64907161822454627203339368991, 6.73777162266986931519532680859, 6.83494534823652798190123119286, 7.02030675873192717083941645936, 7.06480837231137481220671639290, 7.76117048298562390801954091994, 7.82079323609303669575528495344, 8.124051740540613518143885596469
