| L(s) = 1 | + 4·2-s − 4·3-s + 8·4-s − 16·6-s + 8·8-s + 8·9-s + 12·11-s − 32·12-s − 4·16-s − 8·17-s + 32·18-s + 48·22-s − 32·24-s − 8·27-s − 32·32-s − 48·33-s − 32·34-s + 64·36-s + 12·41-s + 24·43-s + 96·44-s + 16·48-s + 32·51-s − 32·54-s − 64·64-s − 192·66-s + 12·67-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 4·4-s − 6.53·6-s + 2.82·8-s + 8/3·9-s + 3.61·11-s − 9.23·12-s − 16-s − 1.94·17-s + 7.54·18-s + 10.2·22-s − 6.53·24-s − 1.53·27-s − 5.65·32-s − 8.35·33-s − 5.48·34-s + 32/3·36-s + 1.87·41-s + 3.65·43-s + 14.4·44-s + 2.30·48-s + 4.48·51-s − 4.35·54-s − 8·64-s − 23.6·66-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220715367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220715367\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 8 T + 32 T^{2} - 16 T^{3} - 353 T^{4} - 16 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 12 T + 72 T^{2} + 744 T^{3} - 8953 T^{4} + 744 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 24 T + 288 T^{2} + 3408 T^{3} + 35567 T^{4} + 3408 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 632 T^{3} - 8153 T^{4} - 632 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 146 T^{2} + 13395 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.222388585541298374123273188096, −9.063529133533466490892419008060, −8.610012821958751638922738443921, −8.605501544872248045331926772498, −7.66520205942067832575048690576, −7.47121923904750975706704075665, −7.21284252878525190772182374626, −6.64848988625281425023659844989, −6.64217966355301426623584171156, −6.57068787004963010496849724737, −6.16520275082593948679464711423, −6.02673628810935109956571652206, −5.67084716115985473628009867567, −5.62786658191637446009706905229, −5.14009866411315656824988647607, −4.62490756329676464249240048063, −4.46334661418564516802034164156, −4.26240099314594303127997269958, −4.19639284153274565464395738441, −3.61167382106582127343927709295, −3.57943967691521696209915189971, −2.55096503201122438505119870413, −2.52182002874068447298479444649, −1.63869030610064386358175872273, −0.879868047995396952604262772193,
0.879868047995396952604262772193, 1.63869030610064386358175872273, 2.52182002874068447298479444649, 2.55096503201122438505119870413, 3.57943967691521696209915189971, 3.61167382106582127343927709295, 4.19639284153274565464395738441, 4.26240099314594303127997269958, 4.46334661418564516802034164156, 4.62490756329676464249240048063, 5.14009866411315656824988647607, 5.62786658191637446009706905229, 5.67084716115985473628009867567, 6.02673628810935109956571652206, 6.16520275082593948679464711423, 6.57068787004963010496849724737, 6.64217966355301426623584171156, 6.64848988625281425023659844989, 7.21284252878525190772182374626, 7.47121923904750975706704075665, 7.66520205942067832575048690576, 8.605501544872248045331926772498, 8.610012821958751638922738443921, 9.063529133533466490892419008060, 9.222388585541298374123273188096
