| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 2·5-s − 8·6-s + 2·7-s + 2·8-s + 6·9-s + 4·10-s − 4·11-s + 4·12-s − 4·14-s − 8·15-s − 4·16-s + 8·17-s − 12·18-s + 20·19-s − 2·20-s + 8·21-s + 8·22-s + 2·23-s + 8·24-s + 5·25-s − 4·27-s + 2·28-s + 4·29-s + 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s − 0.894·5-s − 3.26·6-s + 0.755·7-s + 0.707·8-s + 2·9-s + 1.26·10-s − 1.20·11-s + 1.15·12-s − 1.06·14-s − 2.06·15-s − 16-s + 1.94·17-s − 2.82·18-s + 4.58·19-s − 0.447·20-s + 1.74·21-s + 1.70·22-s + 0.417·23-s + 1.63·24-s + 25-s − 0.769·27-s + 0.377·28-s + 0.742·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040780017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040780017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−9.709702493442751698125376658079, −9.566166560568983836561329147333, −9.491111726959563972177591639996, −8.934326664491490091346343189615, −8.781712046802330974465872367532, −8.233924276768020124611546630617, −8.148890594196736782438973583706, −7.998598881879335966217129888521, −7.82524293043590474923847189075, −7.47418572570541941636517090938, −7.44229396240704930344692185736, −7.08531890079274455098805681232, −6.73378771462966882349478729921, −5.79538136237070114289267443347, −5.46345688450125332645148071140, −5.40559281042754067551752899743, −5.05631759602413752567637533827, −4.56265206948032817000657907568, −3.96152791550059179482244910907, −3.57209220922600236818641414030, −3.12010981632213371051047079961, −2.95395167591308361300504164103, −2.79404569706707804569198721999, −1.58132965029874222363763505206, −1.28371195834214731635635577479,
1.28371195834214731635635577479, 1.58132965029874222363763505206, 2.79404569706707804569198721999, 2.95395167591308361300504164103, 3.12010981632213371051047079961, 3.57209220922600236818641414030, 3.96152791550059179482244910907, 4.56265206948032817000657907568, 5.05631759602413752567637533827, 5.40559281042754067551752899743, 5.46345688450125332645148071140, 5.79538136237070114289267443347, 6.73378771462966882349478729921, 7.08531890079274455098805681232, 7.44229396240704930344692185736, 7.47418572570541941636517090938, 7.82524293043590474923847189075, 7.998598881879335966217129888521, 8.148890594196736782438973583706, 8.233924276768020124611546630617, 8.781712046802330974465872367532, 8.934326664491490091346343189615, 9.491111726959563972177591639996, 9.566166560568983836561329147333, 9.709702493442751698125376658079
