| L(s) = 1 | + 4·3-s − 12·7-s + 8·9-s + 16·13-s − 48·21-s − 8·25-s + 12·27-s − 24·31-s + 8·37-s + 64·39-s + 24·43-s + 72·49-s + 24·61-s − 96·63-s − 20·73-s − 32·75-s + 23·81-s − 192·91-s − 96·93-s − 20·97-s + 12·103-s + 32·111-s + 128·117-s + 8·121-s + 127-s + 96·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4.53·7-s + 8/3·9-s + 4.43·13-s − 10.4·21-s − 8/5·25-s + 2.30·27-s − 4.31·31-s + 1.31·37-s + 10.2·39-s + 3.65·43-s + 72/7·49-s + 3.07·61-s − 12.0·63-s − 2.34·73-s − 3.69·75-s + 23/9·81-s − 20.1·91-s − 9.95·93-s − 2.03·97-s + 1.18·103-s + 3.03·111-s + 11.8·117-s + 8/11·121-s + 0.0887·127-s + 8.45·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.631058124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.631058124\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 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 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−7.21882941327888256406858854487, −6.87767047453469128926400276933, −6.76966120296393570104665420764, −6.62826075166578478662473603672, −6.23455631638341328741887238318, −5.97309433578686554779372364654, −5.90926478266370462989078415274, −5.85424792360133125057835830602, −5.56348688227006660243153311328, −5.45704910753024725401134782941, −4.64707340392894239388772660273, −4.13490424023523478376184179693, −3.96348929002292576271643311653, −3.85850916374785836390468947240, −3.79801806628595102645844559849, −3.57291182840820787735519729547, −3.33969957863949245298513156681, −3.13842988822093115157566073242, −2.75670639595133415185613565284, −2.70076205535659613484145516909, −2.28317104324599484769650277545, −1.68327855821624396119253193632, −1.65470950003202506112875733631, −0.77536610629000277239181481738, −0.53415062381479163906272182814,
0.53415062381479163906272182814, 0.77536610629000277239181481738, 1.65470950003202506112875733631, 1.68327855821624396119253193632, 2.28317104324599484769650277545, 2.70076205535659613484145516909, 2.75670639595133415185613565284, 3.13842988822093115157566073242, 3.33969957863949245298513156681, 3.57291182840820787735519729547, 3.79801806628595102645844559849, 3.85850916374785836390468947240, 3.96348929002292576271643311653, 4.13490424023523478376184179693, 4.64707340392894239388772660273, 5.45704910753024725401134782941, 5.56348688227006660243153311328, 5.85424792360133125057835830602, 5.90926478266370462989078415274, 5.97309433578686554779372364654, 6.23455631638341328741887238318, 6.62826075166578478662473603672, 6.76966120296393570104665420764, 6.87767047453469128926400276933, 7.21882941327888256406858854487
