| L(s) = 1 | + 4·5-s − 4·7-s − 4·11-s − 4·13-s − 2·17-s + 4·19-s + 2·23-s + 10·25-s − 2·29-s − 8·31-s − 16·35-s − 6·37-s − 4·41-s − 2·47-s − 8·49-s + 10·53-s − 16·55-s − 16·59-s + 4·61-s − 16·65-s − 18·67-s − 14·71-s − 2·73-s + 16·77-s − 8·79-s − 6·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.417·23-s + 2·25-s − 0.371·29-s − 1.43·31-s − 2.70·35-s − 0.986·37-s − 0.624·41-s − 0.291·47-s − 8/7·49-s + 1.37·53-s − 2.15·55-s − 2.08·59-s + 0.512·61-s − 1.98·65-s − 2.19·67-s − 1.66·71-s − 0.234·73-s + 1.82·77-s − 0.900·79-s − 0.658·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 4 T + 24 T^{2} + 10 p T^{3} + 250 T^{4} + 10 p^{2} T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 4 T + 34 T^{2} + 118 T^{3} + 502 T^{4} + 118 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 4 T + 48 T^{2} + 138 T^{3} + 918 T^{4} + 138 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 40 T^{2} + 122 T^{3} + 790 T^{4} + 122 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 40 T^{2} - 246 T^{3} + 750 T^{4} - 246 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 2 T + 90 T^{2} + 208 T^{3} + 3526 T^{4} + 208 p T^{5} + 90 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 8 T + 84 T^{2} + 324 T^{3} + 2630 T^{4} + 324 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 6 T + 108 T^{2} + 436 T^{3} + 5318 T^{4} + 436 p T^{5} + 108 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 70 T^{2} + 114 T^{3} + 3226 T^{4} + 114 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 64 T^{2} - 2 T^{3} + 4702 T^{4} - 2 p T^{5} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 136 T^{2} + 390 T^{3} + 8334 T^{4} + 390 p T^{5} + 136 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 10 T + 116 T^{2} - 986 T^{3} + 7230 T^{4} - 986 p T^{5} + 116 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 16 T + 148 T^{2} + 960 T^{3} + 7606 T^{4} + 960 p T^{5} + 148 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 4 T + 104 T^{2} - 16 T^{3} + 4774 T^{4} - 16 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 18 T + 176 T^{2} + 458 T^{3} + 238 T^{4} + 458 p T^{5} + 176 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 14 T + 248 T^{2} + 2478 T^{3} + 24270 T^{4} + 2478 p T^{5} + 248 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 2 T + 84 T^{2} + 710 T^{3} + 4198 T^{4} + 710 p T^{5} + 84 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8 T + 228 T^{2} + 2040 T^{3} + 23478 T^{4} + 2040 p T^{5} + 228 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 6 T + 64 T^{2} + 770 T^{3} + 14334 T^{4} + 770 p T^{5} + 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 6 T + 206 T^{2} + 1232 T^{3} + 20610 T^{4} + 1232 p T^{5} + 206 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 10 T + 172 T^{2} + 120 T^{3} + 5482 T^{4} + 120 p T^{5} + 172 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.07306524635276251495280028732, −5.62205565703813515738413152878, −5.51301197274772717256304113501, −5.46122116586187240174251300461, −5.42064141768733433072613206902, −4.95785515021067753991275516584, −4.88188640209788410149622188068, −4.83662007157765174790773123120, −4.74320768417022004145577455808, −4.14737541600764684638176693088, −3.97346500738625953765478879850, −3.92907613086900503845447591843, −3.77031211946358049397173134723, −3.21856053539771664144718154269, −3.14192053696467007120363339709, −3.08392053920908656973306655200, −2.84702555681576974940341672385, −2.64793796952092717032653780528, −2.40108619857595633889353966534, −2.20093411632297404841970027434, −2.16094053243082541921648854991, −1.48240704567370187585823322923, −1.43668135471966381394200576836, −1.25457106469569294719990132059, −1.22730090590066155035237573194, 0, 0, 0, 0,
1.22730090590066155035237573194, 1.25457106469569294719990132059, 1.43668135471966381394200576836, 1.48240704567370187585823322923, 2.16094053243082541921648854991, 2.20093411632297404841970027434, 2.40108619857595633889353966534, 2.64793796952092717032653780528, 2.84702555681576974940341672385, 3.08392053920908656973306655200, 3.14192053696467007120363339709, 3.21856053539771664144718154269, 3.77031211946358049397173134723, 3.92907613086900503845447591843, 3.97346500738625953765478879850, 4.14737541600764684638176693088, 4.74320768417022004145577455808, 4.83662007157765174790773123120, 4.88188640209788410149622188068, 4.95785515021067753991275516584, 5.42064141768733433072613206902, 5.46122116586187240174251300461, 5.51301197274772717256304113501, 5.62205565703813515738413152878, 6.07306524635276251495280028732
