| L(s) = 1 | − 5·4-s + 10·7-s + 5·9-s − 20·11-s + 9·16-s − 30·17-s − 12·19-s − 70·23-s − 50·28-s − 25·36-s + 40·43-s + 100·44-s − 20·47-s − 23·49-s − 80·61-s + 50·63-s + 35·64-s + 150·68-s − 210·73-s + 60·76-s − 200·77-s − 56·81-s + 80·83-s + 350·92-s − 100·99-s + 90·112-s − 300·119-s + ⋯ |
| L(s) = 1 | − 5/4·4-s + 10/7·7-s + 5/9·9-s − 1.81·11-s + 9/16·16-s − 1.76·17-s − 0.631·19-s − 3.04·23-s − 1.78·28-s − 0.694·36-s + 0.930·43-s + 2.27·44-s − 0.425·47-s − 0.469·49-s − 1.31·61-s + 0.793·63-s + 0.546·64-s + 2.20·68-s − 2.87·73-s + 0.789·76-s − 2.59·77-s − 0.691·81-s + 0.963·83-s + 3.80·92-s − 1.01·99-s + 0.803·112-s − 2.52·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2650344200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2650344200\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 12 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{4} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1357 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2270 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 115 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6637 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7405 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 105 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
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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
−10.86705634616638408814208592843, −10.50396924974894349061832630969, −10.24698051286525255142700647459, −9.694887935871901280984487236939, −9.188352421378282561183553718204, −8.688811452020846393661972312863, −8.277385407913304338628207724717, −7.86198367237999456135358617414, −7.76794793911470860274076314771, −6.95032335826786165132171790800, −6.27463191768634293377222781043, −5.73695616801405915833295792342, −5.22851241826842432970156448305, −4.62329161509123409735729978492, −4.30598937513863677349733840777, −4.13193360861370425539908580404, −2.94804569436370775436398590938, −2.10095601710012672873327671966, −1.75019018244734823419194157173, −0.20493198409441634789163732938,
0.20493198409441634789163732938, 1.75019018244734823419194157173, 2.10095601710012672873327671966, 2.94804569436370775436398590938, 4.13193360861370425539908580404, 4.30598937513863677349733840777, 4.62329161509123409735729978492, 5.22851241826842432970156448305, 5.73695616801405915833295792342, 6.27463191768634293377222781043, 6.95032335826786165132171790800, 7.76794793911470860274076314771, 7.86198367237999456135358617414, 8.277385407913304338628207724717, 8.688811452020846393661972312863, 9.188352421378282561183553718204, 9.694887935871901280984487236939, 10.24698051286525255142700647459, 10.50396924974894349061832630969, 10.86705634616638408814208592843
