| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 8·5-s − 4·6-s + 2·8-s + 2·9-s + 16·10-s − 4·11-s − 4·12-s + 8·13-s − 16·15-s + 3·16-s + 4·17-s + 4·18-s − 4·19-s + 16·20-s − 8·22-s − 10·23-s − 4·24-s + 38·25-s + 16·26-s − 4·27-s − 32·30-s + 8·32-s + 8·33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 3.57·5-s − 1.63·6-s + 0.707·8-s + 2/3·9-s + 5.05·10-s − 1.20·11-s − 1.15·12-s + 2.21·13-s − 4.13·15-s + 3/4·16-s + 0.970·17-s + 0.942·18-s − 0.917·19-s + 3.57·20-s − 1.70·22-s − 2.08·23-s − 0.816·24-s + 38/5·25-s + 3.13·26-s − 0.769·27-s − 5.84·30-s + 1.41·32-s + 1.39·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.416763130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.416763130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 7 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 52 T^{3} + 322 T^{4} - 52 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 220 T^{3} + 967 T^{4} + 220 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2310 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 492 T^{3} + 3326 T^{4} + 492 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 13467 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 98 T^{2} - 756 T^{3} + 5663 T^{4} - 756 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 26978 T^{4} - 2904 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29082 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 140 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.083253607110560605035221046292, −8.315101571278314009171164222710, −8.314516455249807611112829266603, −8.121209127355723559965411973524, −7.74879582059687386815954820889, −7.60527673646873828448902392290, −6.63161469888868005144318806171, −6.55640945911433401334696906741, −6.51121533604265277856010048057, −6.34138717373460538339647393335, −6.13734642520225349004168248340, −5.74305789483629687200663298285, −5.55693282449691795459324791501, −5.26110386334412962193823273798, −5.23106126221524058469644527512, −4.71063462798713931617043135866, −4.68767121262709298304507902618, −4.00102533150305548683968791296, −3.51376026985508116631363542343, −3.44059216215252064596014701302, −2.84979098402822344416459075058, −2.25563626249003951414373620657, −2.10052764842929596527503775915, −1.58110363594391484536567868864, −1.23663971919220932039438123941,
1.23663971919220932039438123941, 1.58110363594391484536567868864, 2.10052764842929596527503775915, 2.25563626249003951414373620657, 2.84979098402822344416459075058, 3.44059216215252064596014701302, 3.51376026985508116631363542343, 4.00102533150305548683968791296, 4.68767121262709298304507902618, 4.71063462798713931617043135866, 5.23106126221524058469644527512, 5.26110386334412962193823273798, 5.55693282449691795459324791501, 5.74305789483629687200663298285, 6.13734642520225349004168248340, 6.34138717373460538339647393335, 6.51121533604265277856010048057, 6.55640945911433401334696906741, 6.63161469888868005144318806171, 7.60527673646873828448902392290, 7.74879582059687386815954820889, 8.121209127355723559965411973524, 8.314516455249807611112829266603, 8.315101571278314009171164222710, 9.083253607110560605035221046292
