| L(s) = 1 | − 3·4-s + 10·13-s + 5·16-s − 4·19-s + 12·31-s − 10·37-s − 20·43-s − 14·49-s − 30·52-s − 22·61-s − 3·64-s − 20·73-s + 12·76-s + 24·79-s − 10·97-s + 20·103-s + 4·109-s + 3·121-s − 36·124-s + 127-s + 131-s + 137-s + 139-s + 30·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 2.77·13-s + 5/4·16-s − 0.917·19-s + 2.15·31-s − 1.64·37-s − 3.04·43-s − 2·49-s − 4.16·52-s − 2.81·61-s − 3/8·64-s − 2.34·73-s + 1.37·76-s + 2.70·79-s − 1.01·97-s + 1.97·103-s + 0.383·109-s + 3/11·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.46·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−8.470229778473156859055797250625, −8.110625633965123632872532116674, −7.72941864643843111924093742058, −6.65161085258324315501961849103, −6.47087996746914035408335230406, −6.14513598061725646029734075717, −5.45093501870232524860958154077, −4.84886591033793439685261848050, −4.56767724494133603201732869283, −3.97564733739679917868066949463, −3.35142569623968027183576325655, −3.18705877871324552307068063652, −1.76125584480235740498550240381, −1.22726947298899977457345127093, 0,
1.22726947298899977457345127093, 1.76125584480235740498550240381, 3.18705877871324552307068063652, 3.35142569623968027183576325655, 3.97564733739679917868066949463, 4.56767724494133603201732869283, 4.84886591033793439685261848050, 5.45093501870232524860958154077, 6.14513598061725646029734075717, 6.47087996746914035408335230406, 6.65161085258324315501961849103, 7.72941864643843111924093742058, 8.110625633965123632872532116674, 8.470229778473156859055797250625
