| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 2·19-s + 4·31-s + 2·36-s − 2·61-s − 4·64-s − 4·76-s + 2·79-s + 81-s + 2·109-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 2·19-s + 4·31-s + 2·36-s − 2·61-s − 4·64-s − 4·76-s + 2·79-s + 81-s + 2·109-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9047571002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9047571002\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−8.900081308112407595395009617072, −8.587397658492804286930549872809, −8.180876631442657523919521986530, −7.85131666767761216617403301025, −7.74600333965871448060914543116, −7.18217574779151980930956789464, −6.39454161943580176961126563383, −6.36265528266144886304107768934, −5.66531677893212505412912027462, −5.60724870331242032306619239667, −4.86721658902835951064928244183, −4.85976528772197147373345262239, −4.45580513022789101715444604662, −3.95945095072030234037523530600, −3.36205106240757318182810814675, −3.02106119192141295114656092404, −2.87805787564232177742981803602, −1.90924684473837826216842209565, −0.969428283997638810696401301606, −0.77875216243639538794843234557,
0.77875216243639538794843234557, 0.969428283997638810696401301606, 1.90924684473837826216842209565, 2.87805787564232177742981803602, 3.02106119192141295114656092404, 3.36205106240757318182810814675, 3.95945095072030234037523530600, 4.45580513022789101715444604662, 4.85976528772197147373345262239, 4.86721658902835951064928244183, 5.60724870331242032306619239667, 5.66531677893212505412912027462, 6.36265528266144886304107768934, 6.39454161943580176961126563383, 7.18217574779151980930956789464, 7.74600333965871448060914543116, 7.85131666767761216617403301025, 8.180876631442657523919521986530, 8.587397658492804286930549872809, 8.900081308112407595395009617072
