| 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
−9.390308195248213925262890437657, −9.296213617157272403992900715524, −8.715467295265622491630819838571, −8.142958304425255316228906608291, −7.87073308765623355773912108302, −7.00703267673792101031355011290, −6.50669597987628232679925335373, −6.13818345852553580176885032227, −5.39084355385318890088281032417, −4.40814951700777868244446600572, −4.05028937660463924524384788233, −3.37337138113178617546598395794, −2.67938102376015495844193066401, −2.04426910526792620981111038824, 0,
2.04426910526792620981111038824, 2.67938102376015495844193066401, 3.37337138113178617546598395794, 4.05028937660463924524384788233, 4.40814951700777868244446600572, 5.39084355385318890088281032417, 6.13818345852553580176885032227, 6.50669597987628232679925335373, 7.00703267673792101031355011290, 7.87073308765623355773912108302, 8.142958304425255316228906608291, 8.715467295265622491630819838571, 9.296213617157272403992900715524, 9.390308195248213925262890437657
