| L(s) = 1 | + 2-s − 2·3-s + 3·4-s + 2·5-s − 2·6-s + 12·7-s + 4·8-s + 9-s + 2·10-s + 8·11-s − 6·12-s − 2·13-s + 12·14-s − 4·15-s + 6·16-s − 2·17-s + 18-s − 4·19-s + 6·20-s − 24·21-s + 8·22-s − 4·23-s − 8·24-s + 25-s − 2·26-s + 2·27-s + 36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 0.816·6-s + 4.53·7-s + 1.41·8-s + 1/3·9-s + 0.632·10-s + 2.41·11-s − 1.73·12-s − 0.554·13-s + 3.20·14-s − 1.03·15-s + 3/2·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 1.34·20-s − 5.23·21-s + 1.70·22-s − 0.834·23-s − 1.63·24-s + 1/5·25-s − 0.392·26-s + 0.384·27-s + 6.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.988401272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.988401272\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T - p T^{2} + T^{3} + 3 T^{4} + p T^{5} - p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T - 18 T^{2} - 8 T^{3} + 263 T^{4} - 8 p T^{5} - 18 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 11 T^{2} - 38 T^{3} - 132 T^{4} - 38 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 14 T^{2} - 64 T^{3} + 3 T^{4} - 64 p T^{5} - 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 26 T^{2} - 24 T^{3} + 1959 T^{4} - 24 p T^{5} - 26 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T - 34 T^{2} - 88 T^{3} - 401 T^{4} - 88 p T^{5} - 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 34 T^{2} - 200 T^{3} - 1793 T^{4} - 200 p T^{5} + 34 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 41 T^{2} + 88 T^{3} + 5808 T^{4} + 88 p T^{5} - 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4\times C_2$ | \( 1 + 16 T + 90 T^{2} + 704 T^{3} + 8219 T^{4} + 704 p T^{5} + 90 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 126 T^{2} + 8 T^{3} + 12143 T^{4} + 8 p T^{5} - 126 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 14 T + 10 T^{2} + 616 T^{3} + 15639 T^{4} + 616 p T^{5} + 10 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 66 T^{2} - 704 T^{3} + 12083 T^{4} - 704 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 16 T + 34 T^{2} + 704 T^{3} + 18579 T^{4} + 704 p T^{5} + 34 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.749815494998566681269278871448, −8.341829321813587286531815393087, −7.937809060689194333426282631233, −7.76171524697917873060628262071, −7.58030126994676825341097694661, −7.33675388557218409967199136868, −6.82351783077576913782327627009, −6.81405122875725009642083286481, −6.57934841499946031168061867119, −6.11064985672256352693027405920, −5.98791580432462128326242566283, −5.55909322785031091588682717259, −5.19541254785176194030133219069, −5.18746264403366801942128825350, −4.93880913473845896716636047058, −4.62904125073879323897800607397, −4.35420026792377673070919680981, −3.85740204545013001073642925246, −3.83766610490813353849045149359, −3.23842223531826032940848586930, −2.38858824201313477038787882866, −1.91356402865840194377750613819, −1.80962384155512714388489781537, −1.63366828595662991941589548408, −1.41040803795204490561101329142,
1.41040803795204490561101329142, 1.63366828595662991941589548408, 1.80962384155512714388489781537, 1.91356402865840194377750613819, 2.38858824201313477038787882866, 3.23842223531826032940848586930, 3.83766610490813353849045149359, 3.85740204545013001073642925246, 4.35420026792377673070919680981, 4.62904125073879323897800607397, 4.93880913473845896716636047058, 5.18746264403366801942128825350, 5.19541254785176194030133219069, 5.55909322785031091588682717259, 5.98791580432462128326242566283, 6.11064985672256352693027405920, 6.57934841499946031168061867119, 6.81405122875725009642083286481, 6.82351783077576913782327627009, 7.33675388557218409967199136868, 7.58030126994676825341097694661, 7.76171524697917873060628262071, 7.937809060689194333426282631233, 8.341829321813587286531815393087, 8.749815494998566681269278871448
