L(s) = 1 | + 2·5-s + 6·9-s + 6·13-s + 16·17-s + 11·25-s − 2·29-s − 32·37-s − 10·41-s + 12·45-s + 11·49-s − 32·53-s + 14·61-s + 12·65-s − 48·73-s + 27·81-s + 32·85-s − 16·89-s + 6·97-s + 26·101-s + 2·113-s + 36·117-s + 19·121-s + 38·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 2·9-s + 1.66·13-s + 3.88·17-s + 11/5·25-s − 0.371·29-s − 5.26·37-s − 1.56·41-s + 1.78·45-s + 11/7·49-s − 4.39·53-s + 1.79·61-s + 1.48·65-s − 5.61·73-s + 3·81-s + 3.47·85-s − 1.69·89-s + 0.609·97-s + 2.58·101-s + 0.188·113-s + 3.32·117-s + 1.72·121-s + 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.628286680\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.628286680\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - 19 T^{2} + 240 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 29 T^{2} + 312 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 35 T^{2} + 264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11 T^{2} - 1728 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 53 T^{2} + 600 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 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 | $C_2^3$ | \( 1 - 59 T^{2} - 1008 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 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 | $C_2^3$ | \( 1 - 91 T^{2} + 1392 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.727328208055644810912531474251, −8.425319170336541108963226923956, −7.968035292621158324564347513473, −7.88768642996580561781338385424, −7.59880852691619146494012319413, −7.09287577731429003498621600581, −7.05762266668823090333220498517, −6.88519037871416508291482998912, −6.69708278865877694783776656619, −6.14854448922832309968041287027, −5.78700925453042974407003725262, −5.76447037055199073739771205970, −5.43407229370130497301127564280, −5.10747130589873339683080282226, −4.90421864001666087939272942155, −4.43851920392037031611869719722, −4.27994393804243794845438353251, −3.53905071799655423922912821965, −3.29732349864558400410302549750, −3.25279673695194443051173092041, −3.23941424425455232015458113527, −2.04491374239691159560657508922, −1.62855886492892848033720070546, −1.41696961880713688107721738451, −1.15803279730477766421113583115,
1.15803279730477766421113583115, 1.41696961880713688107721738451, 1.62855886492892848033720070546, 2.04491374239691159560657508922, 3.23941424425455232015458113527, 3.25279673695194443051173092041, 3.29732349864558400410302549750, 3.53905071799655423922912821965, 4.27994393804243794845438353251, 4.43851920392037031611869719722, 4.90421864001666087939272942155, 5.10747130589873339683080282226, 5.43407229370130497301127564280, 5.76447037055199073739771205970, 5.78700925453042974407003725262, 6.14854448922832309968041287027, 6.69708278865877694783776656619, 6.88519037871416508291482998912, 7.05762266668823090333220498517, 7.09287577731429003498621600581, 7.59880852691619146494012319413, 7.88768642996580561781338385424, 7.968035292621158324564347513473, 8.425319170336541108963226923956, 8.727328208055644810912531474251