| L(s) = 1 | − 2·2-s + 4·3-s − 4-s − 8·6-s + 10·7-s + 4·8-s + 11·9-s + 2·11-s − 4·12-s + 8·13-s − 20·14-s − 8·17-s − 22·18-s − 2·19-s + 40·21-s − 4·22-s − 14·23-s + 16·24-s − 16·26-s + 20·27-s − 10·28-s − 12·31-s − 2·32-s + 8·33-s + 16·34-s − 11·36-s + 12·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s − 1/2·4-s − 3.26·6-s + 3.77·7-s + 1.41·8-s + 11/3·9-s + 0.603·11-s − 1.15·12-s + 2.21·13-s − 5.34·14-s − 1.94·17-s − 5.18·18-s − 0.458·19-s + 8.72·21-s − 0.852·22-s − 2.91·23-s + 3.26·24-s − 3.13·26-s + 3.84·27-s − 1.88·28-s − 2.15·31-s − 0.353·32-s + 1.39·33-s + 2.74·34-s − 1.83·36-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779956473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779956473\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 4 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 226 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 288 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 672 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 137 T^{2} + 1224 T^{3} + 11492 T^{4} + 1224 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 144 T^{2} - 600 T^{3} + 10991 T^{4} - 600 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 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
−9.359998870139710593502553519480, −8.724633377456072125069331342388, −8.712706971656801701001470968287, −8.571087086125865018280357885223, −8.514992574883318586313382205631, −7.992302096173380204486950388917, −7.87213923219163673730412042619, −7.75289709037691972528878450711, −7.74493522192712570841171594213, −7.03387974945067847873161206004, −6.83254941459970093343343053223, −6.17686119981450019069870709037, −5.94698876046510934285152931536, −5.74758904399241084903752998965, −4.97703755259523160912737068799, −4.83674537912692526457307710184, −4.26756424717461960487827618232, −4.13480834724042785373400456560, −4.09170186498259751481065436535, −3.83444985962074491034512330671, −3.03844382634343125327799957583, −2.29619298643033443023553836225, −1.81299987570860903987960866303, −1.79179775446068053194037104892, −1.30726378938097481718546695029,
1.30726378938097481718546695029, 1.79179775446068053194037104892, 1.81299987570860903987960866303, 2.29619298643033443023553836225, 3.03844382634343125327799957583, 3.83444985962074491034512330671, 4.09170186498259751481065436535, 4.13480834724042785373400456560, 4.26756424717461960487827618232, 4.83674537912692526457307710184, 4.97703755259523160912737068799, 5.74758904399241084903752998965, 5.94698876046510934285152931536, 6.17686119981450019069870709037, 6.83254941459970093343343053223, 7.03387974945067847873161206004, 7.74493522192712570841171594213, 7.75289709037691972528878450711, 7.87213923219163673730412042619, 7.992302096173380204486950388917, 8.514992574883318586313382205631, 8.571087086125865018280357885223, 8.712706971656801701001470968287, 8.724633377456072125069331342388, 9.359998870139710593502553519480
