| L(s) = 1 | + 3·3-s + 5-s − 3·7-s + 6·9-s − 2·11-s + 2·13-s + 3·15-s + 8·17-s + 16·19-s − 9·21-s + 3·23-s + 9·27-s + 29-s − 6·33-s − 3·35-s − 8·37-s + 6·39-s − 5·41-s − 8·43-s + 6·45-s + 7·47-s + 7·49-s + 24·51-s − 4·53-s − 2·55-s + 48·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 1.13·7-s + 2·9-s − 0.603·11-s + 0.554·13-s + 0.774·15-s + 1.94·17-s + 3.67·19-s − 1.96·21-s + 0.625·23-s + 1.73·27-s + 0.185·29-s − 1.04·33-s − 0.507·35-s − 1.31·37-s + 0.960·39-s − 0.780·41-s − 1.21·43-s + 0.894·45-s + 1.02·47-s + 49-s + 3.36·51-s − 0.549·53-s − 0.269·55-s + 6.35·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.297630842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.297630842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.18892888898760967169349237883, −10.09150939549746597927053920240, −9.608617443960571825905924154476, −9.547609695683098551353532841257, −8.849943478974440186982724255510, −8.695231987502861283142760352908, −7.82029979437575278300668026496, −7.77650838471059596261502091712, −7.12820048389790173798542229356, −7.10679583974709831986743781916, −6.12156519138280435750732477786, −5.73892550641310864587451372998, −5.17055850751330882699124209379, −4.83168194940643303100738536912, −3.59621252759044902344282648574, −3.51759090327571145232493760843, −2.99728129544353360426983778020, −2.83625030546701733890904303266, −1.60763789755408143790910168679, −1.13874338536850994095405650931,
1.13874338536850994095405650931, 1.60763789755408143790910168679, 2.83625030546701733890904303266, 2.99728129544353360426983778020, 3.51759090327571145232493760843, 3.59621252759044902344282648574, 4.83168194940643303100738536912, 5.17055850751330882699124209379, 5.73892550641310864587451372998, 6.12156519138280435750732477786, 7.10679583974709831986743781916, 7.12820048389790173798542229356, 7.77650838471059596261502091712, 7.82029979437575278300668026496, 8.695231987502861283142760352908, 8.849943478974440186982724255510, 9.547609695683098551353532841257, 9.608617443960571825905924154476, 10.09150939549746597927053920240, 10.18892888898760967169349237883
