| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s + 7-s + 2·8-s − 3·9-s + 3·11-s − 2·12-s − 2·13-s − 2·14-s − 4·16-s + 18·17-s + 6·18-s + 2·19-s − 2·21-s − 6·22-s − 3·23-s − 4·24-s + 4·26-s + 14·27-s + 28-s + 3·29-s + 2·31-s + 2·32-s − 6·33-s − 36·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 0.377·7-s + 0.707·8-s − 9-s + 0.904·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 16-s + 4.36·17-s + 1.41·18-s + 0.458·19-s − 0.436·21-s − 1.27·22-s − 0.625·23-s − 0.816·24-s + 0.784·26-s + 2.69·27-s + 0.188·28-s + 0.557·29-s + 0.359·31-s + 0.353·32-s − 1.04·33-s − 6.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7513276936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7513276936\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + 10 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 18 p T^{5} - 31 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 9 T - 25 T^{2} + 108 T^{3} + 5220 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 18 p T^{5} - 103 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 11 T + 102 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9 T - 97 T^{2} - 108 T^{3} + 18072 T^{4} - 108 p T^{5} - 97 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 11 T - 95 T^{2} + 242 T^{3} + 25510 T^{4} + 242 p T^{5} - 95 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.238907163307847926973934599389, −7.59642553936461887459791423463, −7.58181438181280895180891645024, −7.56168929819908919274144386886, −7.39557845651719156679207722853, −6.72138483076677685519083495070, −6.45727953836589963641012956699, −6.24843277043642241968095167081, −6.01243059635382557420088787259, −5.97225882102372490357664095155, −5.37226378574877034614356017304, −5.36962303761258296357876959444, −5.14529696431151940772261773175, −4.80381608457137654242643483043, −4.39829875938970653391149644738, −4.32873757139216511598232275913, −3.66282152371905287077999733307, −3.26973320707507731294267519279, −3.21436384597287508819489242448, −2.94340361932433208035035069957, −2.37434490956744227691583248713, −1.74274670551797321247400523076, −1.37399023293577192992577820503, −0.795769076564163916437065553995, −0.69330471787648547657972885818,
0.69330471787648547657972885818, 0.795769076564163916437065553995, 1.37399023293577192992577820503, 1.74274670551797321247400523076, 2.37434490956744227691583248713, 2.94340361932433208035035069957, 3.21436384597287508819489242448, 3.26973320707507731294267519279, 3.66282152371905287077999733307, 4.32873757139216511598232275913, 4.39829875938970653391149644738, 4.80381608457137654242643483043, 5.14529696431151940772261773175, 5.36962303761258296357876959444, 5.37226378574877034614356017304, 5.97225882102372490357664095155, 6.01243059635382557420088787259, 6.24843277043642241968095167081, 6.45727953836589963641012956699, 6.72138483076677685519083495070, 7.39557845651719156679207722853, 7.56168929819908919274144386886, 7.58181438181280895180891645024, 7.59642553936461887459791423463, 8.238907163307847926973934599389
