| L(s) = 1 | − 2-s + 6·3-s + 4-s − 6·6-s − 8-s + 21·9-s − 10·11-s + 6·12-s + 16-s − 8·17-s − 21·18-s + 4·19-s + 10·22-s − 6·24-s − 10·25-s + 54·27-s − 32-s − 60·33-s + 8·34-s + 21·36-s − 4·38-s − 18·41-s − 24·43-s − 10·44-s + 6·48-s + 49-s + 10·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.46·3-s + 1/2·4-s − 2.44·6-s − 0.353·8-s + 7·9-s − 3.01·11-s + 1.73·12-s + 1/4·16-s − 1.94·17-s − 4.94·18-s + 0.917·19-s + 2.13·22-s − 1.22·24-s − 2·25-s + 10.3·27-s − 0.176·32-s − 10.4·33-s + 1.37·34-s + 7/2·36-s − 0.648·38-s − 2.81·41-s − 3.65·43-s − 1.50·44-s + 0.866·48-s + 1/7·49-s + 1.41·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−7.965038606433296076845496447561, −7.959201586617154759738546904094, −7.28962231264252311819764478366, −6.90516587374622685438056526369, −6.54235846854329487226858841508, −5.52075155031749515074465096823, −4.88738837750963504497893956492, −4.71388868426411291420513942193, −3.68813633084464247818027803186, −3.31592951327620786086829859540, −3.16464702872454555446785806554, −2.29104151969152161829103899523, −2.10681498002416439479097053291, −1.86203973153105577745339857423, 0,
1.86203973153105577745339857423, 2.10681498002416439479097053291, 2.29104151969152161829103899523, 3.16464702872454555446785806554, 3.31592951327620786086829859540, 3.68813633084464247818027803186, 4.71388868426411291420513942193, 4.88738837750963504497893956492, 5.52075155031749515074465096823, 6.54235846854329487226858841508, 6.90516587374622685438056526369, 7.28962231264252311819764478366, 7.959201586617154759738546904094, 7.965038606433296076845496447561
