| L(s) = 1 | − 2·2-s + 4-s + 2·5-s − 2·7-s + 2·8-s − 4·10-s − 8·11-s + 2·13-s + 4·14-s − 4·16-s + 2·20-s + 16·22-s + 2·23-s + 8·25-s − 4·26-s − 2·28-s − 16·29-s − 8·31-s + 2·32-s − 4·35-s − 24·37-s + 4·40-s − 6·41-s + 4·43-s − 8·44-s − 4·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.707·8-s − 1.26·10-s − 2.41·11-s + 0.554·13-s + 1.06·14-s − 16-s + 0.447·20-s + 3.41·22-s + 0.417·23-s + 8/5·25-s − 0.784·26-s − 0.377·28-s − 2.97·29-s − 1.43·31-s + 0.353·32-s − 0.676·35-s − 3.94·37-s + 0.632·40-s − 0.937·41-s + 0.609·43-s − 1.20·44-s − 0.589·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1261688393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1261688393\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T - 4 T^{2} + 4 T^{3} + 19 T^{4} + 4 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 22 T^{3} - 68 T^{4} - 22 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 29 T^{2} + 104 T^{3} + 400 T^{4} + 104 p T^{5} + 29 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 16 T + 137 T^{2} + 976 T^{3} + 5896 T^{4} + 976 p T^{5} + 137 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 52 T^{2} + 36 T^{3} + 4587 T^{4} + 36 p T^{5} - 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 26 T^{2} + 176 T^{3} - 773 T^{4} + 176 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 34 T^{2} + 32 T^{3} + 4387 T^{4} + 32 p T^{5} - 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T + 41 T^{2} - 572 T^{3} - 3800 T^{4} - 572 p T^{5} + 41 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T - 83 T^{2} + 92 T^{3} + 5104 T^{4} + 92 p T^{5} - 83 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 14 T^{2} + 416 T^{3} - 4661 T^{4} + 416 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 14 T + 179 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 17 T^{2} - 468 T^{3} - 2712 T^{4} - 468 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 232 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 22 T + 172 T^{2} + 2596 T^{3} + 40987 T^{4} + 2596 p T^{5} + 172 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.50343023347658297004144691260, −6.45067596669846458275625739340, −5.94739075853214764131726926404, −5.68940830979196286181444393927, −5.49933495030901630931239670510, −5.44405449424575371312589533902, −5.41890917998285990853110665142, −5.24596935984901056438058120416, −4.77445412032991310138170276528, −4.70079091388146782023133003278, −4.31383190499627625342435877829, −4.02794798507960119317426867884, −3.76119310597636983660811651881, −3.53610977633730726269567666634, −3.41479546194235440042794125489, −2.98283677983013092562436453214, −2.80516177645499214022127405530, −2.52979534827778185615574596786, −2.44433801294175895821029272154, −1.76899579406743106525331365008, −1.61189512844226034472100702988, −1.55348797070545485839239480369, −1.29793889288221724067249575558, −0.24224900280989825176447733712, −0.23210311593268611110266612020,
0.23210311593268611110266612020, 0.24224900280989825176447733712, 1.29793889288221724067249575558, 1.55348797070545485839239480369, 1.61189512844226034472100702988, 1.76899579406743106525331365008, 2.44433801294175895821029272154, 2.52979534827778185615574596786, 2.80516177645499214022127405530, 2.98283677983013092562436453214, 3.41479546194235440042794125489, 3.53610977633730726269567666634, 3.76119310597636983660811651881, 4.02794798507960119317426867884, 4.31383190499627625342435877829, 4.70079091388146782023133003278, 4.77445412032991310138170276528, 5.24596935984901056438058120416, 5.41890917998285990853110665142, 5.44405449424575371312589533902, 5.49933495030901630931239670510, 5.68940830979196286181444393927, 5.94739075853214764131726926404, 6.45067596669846458275625739340, 6.50343023347658297004144691260
