| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s − 2·5-s − 8·6-s + 20·8-s + 3·9-s − 8·10-s − 4·11-s − 20·12-s + 4·15-s + 35·16-s − 4·17-s + 12·18-s − 10·19-s − 20·20-s − 16·22-s + 2·23-s − 40·24-s + 5·25-s − 10·27-s + 4·29-s + 16·30-s + 24·31-s + 56·32-s + 8·33-s − 16·34-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 5·4-s − 0.894·5-s − 3.26·6-s + 7.07·8-s + 9-s − 2.52·10-s − 1.20·11-s − 5.77·12-s + 1.03·15-s + 35/4·16-s − 0.970·17-s + 2.82·18-s − 2.29·19-s − 4.47·20-s − 3.41·22-s + 0.417·23-s − 8.16·24-s + 25-s − 1.92·27-s + 0.742·29-s + 2.92·30-s + 4.31·31-s + 9.89·32-s + 1.39·33-s − 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.23617610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.23617610\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 p T^{3} - 4 p T^{4} - 2 p^{2} T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 43 T^{2} + 10 p T^{3} + 52 p T^{4} + 10 p^{2} T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 368 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 26 T^{2} - 144 T^{3} + 3483 T^{4} - 144 p T^{5} + 26 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 110 T^{2} + 80 T^{3} + 9379 T^{4} + 80 p T^{5} - 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.14538418315273409030869738086, −6.82464746729667166205671043561, −6.77708823834667429875720263524, −6.29295068863253825526834624508, −6.20935706270545946990250286488, −6.20814771907761977505849638444, −6.02658707998982627249246281286, −5.44355491864808982179595425905, −5.40635787501367757965196985665, −4.91579774995962534684418416263, −4.88630363271291972867257963614, −4.81869294264265596085641944901, −4.40647303453820468015884355694, −4.33901387381464265740338154836, −4.09454820991011250437365884225, −3.76309616241093245233201141054, −3.47854167908856265857120431456, −3.22454651869963190409009759413, −2.87919173027555136321973424140, −2.44466757050413933184142312459, −2.35795815038552955639303282708, −2.10919020864582238832907133752, −1.68101488821066779600766274575, −0.76296211125101271423519540637, −0.68360126664365061279217092826,
0.68360126664365061279217092826, 0.76296211125101271423519540637, 1.68101488821066779600766274575, 2.10919020864582238832907133752, 2.35795815038552955639303282708, 2.44466757050413933184142312459, 2.87919173027555136321973424140, 3.22454651869963190409009759413, 3.47854167908856265857120431456, 3.76309616241093245233201141054, 4.09454820991011250437365884225, 4.33901387381464265740338154836, 4.40647303453820468015884355694, 4.81869294264265596085641944901, 4.88630363271291972867257963614, 4.91579774995962534684418416263, 5.40635787501367757965196985665, 5.44355491864808982179595425905, 6.02658707998982627249246281286, 6.20814771907761977505849638444, 6.20935706270545946990250286488, 6.29295068863253825526834624508, 6.77708823834667429875720263524, 6.82464746729667166205671043561, 7.14538418315273409030869738086
