| L(s) = 1 | − 6·3-s − 4·4-s + 11·7-s + 27·9-s + 24·12-s − 22·13-s − 74·19-s − 66·21-s − 25·25-s − 108·27-s − 44·28-s + 13·31-s − 108·36-s − 47·37-s + 132·39-s + 22·43-s + 72·49-s + 88·52-s + 444·57-s − 242·61-s + 297·63-s + 64·64-s − 26·67-s + 46·73-s + 150·75-s + 296·76-s − 11·79-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 11/7·7-s + 3·9-s + 2·12-s − 1.69·13-s − 3.89·19-s − 3.14·21-s − 25-s − 4·27-s − 1.57·28-s + 0.419·31-s − 3·36-s − 1.27·37-s + 3.38·39-s + 0.511·43-s + 1.46·49-s + 1.69·52-s + 7.78·57-s − 3.96·61-s + 33/7·63-s + 64-s − 0.388·67-s + 0.630·73-s + 2·75-s + 3.89·76-s − 0.139·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07634333436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07634333436\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( 1 - 11 T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 22 T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 121 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−12.28098180130650278143803097317, −11.41154205770298627056374137314, −10.89647296126862324065591957790, −10.48872143924438848613087220949, −10.46501484034397229677229290840, −9.549493090494254944408459523508, −9.254732234435825679664916301230, −8.322723816320163164694817054741, −8.228644477180900702749574112310, −7.31213403915285206061041613611, −7.04823044115706988395551066386, −6.16379386611979350133409283999, −5.97545119478761872765213512549, −4.97743196019390974318839308123, −4.87923288939970046330695177580, −4.24021069917953898901604356891, −4.19398571764677280622879680497, −2.19795617908683694191004955252, −1.68425528201422825653393468382, −0.15779426257287457690172367114,
0.15779426257287457690172367114, 1.68425528201422825653393468382, 2.19795617908683694191004955252, 4.19398571764677280622879680497, 4.24021069917953898901604356891, 4.87923288939970046330695177580, 4.97743196019390974318839308123, 5.97545119478761872765213512549, 6.16379386611979350133409283999, 7.04823044115706988395551066386, 7.31213403915285206061041613611, 8.228644477180900702749574112310, 8.322723816320163164694817054741, 9.254732234435825679664916301230, 9.549493090494254944408459523508, 10.46501484034397229677229290840, 10.48872143924438848613087220949, 10.89647296126862324065591957790, 11.41154205770298627056374137314, 12.28098180130650278143803097317
