L(s) = 1 | − 2·3-s + 9-s − 12·11-s − 10·25-s + 4·27-s + 24·33-s − 14·49-s − 12·59-s − 4·73-s + 20·75-s − 11·81-s − 36·83-s + 20·97-s − 12·99-s − 12·107-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 3.61·11-s − 2·25-s + 0.769·27-s + 4.17·33-s − 2·49-s − 1.56·59-s − 0.468·73-s + 2.30·75-s − 1.22·81-s − 3.95·83-s + 2.03·97-s − 1.20·99-s − 1.16·107-s + 7.81·121-s + 0.0887·127-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84589620037637178833013432206, −7.49638267192940041493128754308, −7.30853411341707779435006421610, −6.29051795613585298614624812000, −6.14544538956463126183783806875, −5.43412469058090981748265870176, −5.39674718935401561719719827206, −4.81417604656358256657634712562, −4.43148007247498493089277458199, −3.51187327243886123358753236242, −2.81978937059917029896777831283, −2.49619173311123854201331757855, −1.60129609129764731842047004184, 0, 0,
1.60129609129764731842047004184, 2.49619173311123854201331757855, 2.81978937059917029896777831283, 3.51187327243886123358753236242, 4.43148007247498493089277458199, 4.81417604656358256657634712562, 5.39674718935401561719719827206, 5.43412469058090981748265870176, 6.14544538956463126183783806875, 6.29051795613585298614624812000, 7.30853411341707779435006421610, 7.49638267192940041493128754308, 7.84589620037637178833013432206