| L(s) = 1 | + 2-s − 4·3-s + 5-s − 4·6-s + 7-s + 13·9-s + 10-s − 11-s + 2·13-s + 14-s − 4·15-s + 2·17-s + 13·18-s + 11·19-s − 4·21-s − 22-s + 24·23-s + 2·26-s − 30·27-s + 6·29-s − 4·30-s − 6·31-s − 32-s + 4·33-s + 2·34-s + 35-s + 12·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 0.447·5-s − 1.63·6-s + 0.377·7-s + 13/3·9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 0.485·17-s + 3.06·18-s + 2.52·19-s − 0.872·21-s − 0.213·22-s + 5.00·23-s + 0.392·26-s − 5.77·27-s + 1.11·29-s − 0.730·30-s − 1.07·31-s − 0.176·32-s + 0.696·33-s + 0.342·34-s + 0.169·35-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.902625199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.902625199\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} + 44 p T^{5} - 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 11 T + 27 T^{2} + 137 T^{3} - 1120 T^{4} + 137 p T^{5} + 27 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 6 T - 13 T^{2} + 42 T^{3} + 625 T^{4} + 42 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 6 T - 15 T^{2} - 46 T^{3} + 729 T^{4} - 46 p T^{5} - 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 12 T + 27 T^{2} + 20 T^{3} + 441 T^{4} + 20 p T^{5} + 27 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 20 T + 149 T^{2} + 620 T^{3} + 2861 T^{4} + 620 p T^{5} + 149 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 12 T + 17 T^{2} - 570 T^{3} - 5819 T^{4} - 570 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T - 37 T^{2} + 90 T^{3} + 2401 T^{4} + 90 p T^{5} - 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 9 T - 13 T^{2} + 603 T^{3} - 4160 T^{4} + 603 p T^{5} - 13 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - 11 T + 51 T^{2} - 11 p T^{3} + p^{2} T^{4} )( 1 + 29 T + 331 T^{2} + 29 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_{4}$ | \( ( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 12 T + 73 T^{2} + 1074 T^{3} + 14005 T^{4} + 1074 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 18 T + 71 T^{2} + 1236 T^{3} - 19691 T^{4} + 1236 p T^{5} + 71 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 6 T - 3 T^{2} - 688 T^{3} + 10365 T^{4} - 688 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 15 T + 107 T^{2} + 1515 T^{3} + 20704 T^{4} + 1515 p T^{5} + 107 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 203 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 23 T + 207 T^{2} + 2125 T^{3} + 27176 T^{4} + 2125 p T^{5} + 207 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.38394596277946732537898082954, −6.84834675328297815007934050472, −6.82250623746700134194754160944, −6.81409651013972598465210708964, −6.70953956149675691126402222282, −6.00610576834130099231341737649, −5.94803595470040168534811668855, −5.75174126007722852476292903827, −5.53459122344021600823175703343, −5.02868765053528121284772560271, −4.98741469343264639820943897600, −4.87812660532622234742415692350, −4.78372030970651450076704902417, −4.75652446944338137937298844416, −4.04168758151204427292026511147, −3.80178593189049742422345854352, −3.40902918521480707039789427746, −3.30949161081998482067016729011, −2.88977057114442357074772784731, −2.73416033341834546448367524828, −2.01741920896444142621171316237, −1.44238467357729267598038128890, −1.34295148034909589265862608145, −0.915799072505823807961652277666, −0.75985646841372947604008526393,
0.75985646841372947604008526393, 0.915799072505823807961652277666, 1.34295148034909589265862608145, 1.44238467357729267598038128890, 2.01741920896444142621171316237, 2.73416033341834546448367524828, 2.88977057114442357074772784731, 3.30949161081998482067016729011, 3.40902918521480707039789427746, 3.80178593189049742422345854352, 4.04168758151204427292026511147, 4.75652446944338137937298844416, 4.78372030970651450076704902417, 4.87812660532622234742415692350, 4.98741469343264639820943897600, 5.02868765053528121284772560271, 5.53459122344021600823175703343, 5.75174126007722852476292903827, 5.94803595470040168534811668855, 6.00610576834130099231341737649, 6.70953956149675691126402222282, 6.81409651013972598465210708964, 6.82250623746700134194754160944, 6.84834675328297815007934050472, 7.38394596277946732537898082954
