| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s − 4·8-s + 4·9-s + 8·10-s + 2·11-s − 4·12-s − 16·13-s + 8·15-s + 8·16-s − 10·17-s − 8·18-s − 16·19-s − 8·20-s − 4·22-s − 8·23-s + 8·24-s + 5·25-s + 32·26-s − 4·27-s + 4·29-s − 16·30-s − 8·32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s − 1.41·8-s + 4/3·9-s + 2.52·10-s + 0.603·11-s − 1.15·12-s − 4.43·13-s + 2.06·15-s + 2·16-s − 2.42·17-s − 1.88·18-s − 3.67·19-s − 1.78·20-s − 0.852·22-s − 1.66·23-s + 1.63·24-s + 25-s + 6.27·26-s − 0.769·27-s + 0.742·29-s − 2.92·30-s − 1.41·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 26 T^{3} - 32 T^{4} - 26 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 26 T^{2} - 140 T^{3} - 1169 T^{4} - 140 p T^{5} + 26 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 208 T^{3} - 1604 T^{4} - 208 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 28 T^{3} - 386 T^{4} - 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^3$ | \( 1 + 58 T^{2} + 2403 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 18 T + 205 T^{2} + 1746 T^{3} + 12036 T^{4} + 1746 p T^{5} + 205 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 60 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 89 T^{2} - 392 T^{3} - 6788 T^{4} - 392 p T^{5} + 89 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 18 T + 149 T^{2} + 1242 T^{3} + 10644 T^{4} + 1242 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 22 T + 170 T^{2} + 172 T^{3} - 4433 T^{4} + 172 p T^{5} + 170 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 180 T^{2} - 1596 T^{3} + 15143 T^{4} - 1596 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 22 T^{2} - 144 T^{3} + 747 T^{4} - 144 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 552 T^{3} - 6433 T^{4} + 552 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 382 T^{2} + 4560 T^{3} + 45267 T^{4} + 4560 p T^{5} + 382 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.343716928147225174147977765893, −8.067399786465062070181695323823, −7.62223003290944654281502400379, −7.56145874580873469687865854218, −7.40810214374920158597720733841, −7.00448686627450014830656775618, −6.82349963546447386056145331578, −6.77638260468628345625477246147, −6.38007940162269626272964244612, −6.24038404467639520010822702077, −6.13738793141903282941298804146, −5.61343312881000907332171994833, −5.11743813279477675676768311027, −4.99757898269429124662773666321, −4.67431634669222900362865485025, −4.46945414671155199877642185554, −4.39553268937052852534913235869, −3.99083584656751184105673174570, −3.96970746414288822935619009453, −3.08880807035504414727149734471, −2.97742262229730126814430026119, −2.68639923993882210255018079806, −1.99309395643713419051093599310, −1.81717014995632610045885379768, −1.81535260798041970320295115993, 0, 0, 0, 0,
1.81535260798041970320295115993, 1.81717014995632610045885379768, 1.99309395643713419051093599310, 2.68639923993882210255018079806, 2.97742262229730126814430026119, 3.08880807035504414727149734471, 3.96970746414288822935619009453, 3.99083584656751184105673174570, 4.39553268937052852534913235869, 4.46945414671155199877642185554, 4.67431634669222900362865485025, 4.99757898269429124662773666321, 5.11743813279477675676768311027, 5.61343312881000907332171994833, 6.13738793141903282941298804146, 6.24038404467639520010822702077, 6.38007940162269626272964244612, 6.77638260468628345625477246147, 6.82349963546447386056145331578, 7.00448686627450014830656775618, 7.40810214374920158597720733841, 7.56145874580873469687865854218, 7.62223003290944654281502400379, 8.067399786465062070181695323823, 8.343716928147225174147977765893
