| L(s) = 1 | − 2·3-s − 4-s + 4·7-s + 9-s + 2·12-s − 3·16-s − 8·19-s − 8·21-s − 5·25-s + 4·27-s − 4·28-s − 36-s − 8·37-s + 6·48-s − 2·49-s + 16·57-s + 4·63-s + 7·64-s − 8·73-s + 10·75-s + 8·76-s − 11·81-s + 8·84-s + 16·97-s + 5·100-s − 4·108-s + 16·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.577·12-s − 3/4·16-s − 1.83·19-s − 1.74·21-s − 25-s + 0.769·27-s − 0.755·28-s − 1/6·36-s − 1.31·37-s + 0.866·48-s − 2/7·49-s + 2.11·57-s + 0.503·63-s + 7/8·64-s − 0.936·73-s + 1.15·75-s + 0.917·76-s − 1.22·81-s + 0.872·84-s + 1.62·97-s + 1/2·100-s − 0.384·108-s + 1.51·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.765943762067730233506644822169, −8.926879393574297996153906747429, −8.695372705841464747364612145313, −8.179017934071532057682505257545, −7.65915795055618436127855465882, −6.95271599818545416907011775238, −6.40006853654927336554404315520, −5.92676279224525644903587034681, −5.21879136513047480917226119288, −4.79899508194714834866543797801, −4.40518336224367655451174519155, −3.68219362339604559493338616127, −2.36933666514572285556401057482, −1.59784097653209773151986533836, 0,
1.59784097653209773151986533836, 2.36933666514572285556401057482, 3.68219362339604559493338616127, 4.40518336224367655451174519155, 4.79899508194714834866543797801, 5.21879136513047480917226119288, 5.92676279224525644903587034681, 6.40006853654927336554404315520, 6.95271599818545416907011775238, 7.65915795055618436127855465882, 8.179017934071532057682505257545, 8.695372705841464747364612145313, 8.926879393574297996153906747429, 9.765943762067730233506644822169
