L(s) = 1 | − 2·3-s − 4-s − 4·7-s + 9-s + 2·12-s − 3·13-s − 3·16-s + 6·19-s + 8·21-s − 10·25-s + 4·27-s + 4·28-s − 14·31-s − 36-s + 12·37-s + 6·39-s − 2·43-s + 6·48-s + 9·49-s + 3·52-s − 12·57-s − 20·61-s − 4·63-s + 7·64-s + 2·67-s − 8·73-s + 20·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s − 1.51·7-s + 1/3·9-s + 0.577·12-s − 0.832·13-s − 3/4·16-s + 1.37·19-s + 1.74·21-s − 2·25-s + 0.769·27-s + 0.755·28-s − 2.51·31-s − 1/6·36-s + 1.97·37-s + 0.960·39-s − 0.304·43-s + 0.866·48-s + 9/7·49-s + 0.416·52-s − 1.58·57-s − 2.56·61-s − 0.503·63-s + 7/8·64-s + 0.244·67-s − 0.936·73-s + 2.30·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 18 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.71780362935470178287102279029, −7.19356419594824377526982729317, −6.85131222784513026706605859226, −6.27010423831462867277947279327, −5.87214331264323289428027877582, −5.62054609874974790317624234396, −5.09601624145493560558789090672, −4.58855833631578228050592708255, −3.99338287732795096289832344949, −3.56623646346365020852704538669, −2.89477212382939041166889656695, −2.33926579557529872453744822484, −1.32492343281959828647528256526, 0, 0,
1.32492343281959828647528256526, 2.33926579557529872453744822484, 2.89477212382939041166889656695, 3.56623646346365020852704538669, 3.99338287732795096289832344949, 4.58855833631578228050592708255, 5.09601624145493560558789090672, 5.62054609874974790317624234396, 5.87214331264323289428027877582, 6.27010423831462867277947279327, 6.85131222784513026706605859226, 7.19356419594824377526982729317, 7.71780362935470178287102279029