| L(s) = 1 | + 2·3-s + 9-s + 4·19-s − 10·25-s − 4·27-s − 20·43-s − 14·49-s + 8·57-s − 28·67-s − 4·73-s − 20·75-s − 11·81-s + 20·97-s + 14·121-s + 127-s − 40·129-s + 131-s + 137-s + 139-s − 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 4·171-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.917·19-s − 2·25-s − 0.769·27-s − 3.04·43-s − 2·49-s + 1.05·57-s − 3.42·67-s − 0.468·73-s − 2.30·75-s − 1.22·81-s + 2.03·97-s + 1.27·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.30·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.305·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.337928305679105143801605101388, −7.84589620037637178833013432206, −7.30853411341707779435006421610, −7.14838005963241329022744903293, −6.14544538956463126183783806875, −6.13287781737238367476060833635, −5.39674718935401561719719827206, −4.81417604656358256657634712562, −4.38680180444175422137402538929, −3.51187327243886123358753236242, −3.38746738060370983088123271545, −2.81978937059917029896777831283, −1.90996633779170653420430529631, −1.60129609129764731842047004184, 0,
1.60129609129764731842047004184, 1.90996633779170653420430529631, 2.81978937059917029896777831283, 3.38746738060370983088123271545, 3.51187327243886123358753236242, 4.38680180444175422137402538929, 4.81417604656358256657634712562, 5.39674718935401561719719827206, 6.13287781737238367476060833635, 6.14544538956463126183783806875, 7.14838005963241329022744903293, 7.30853411341707779435006421610, 7.84589620037637178833013432206, 8.337928305679105143801605101388
