| L(s) = 1 | + 2·5-s + 8·19-s + 12·23-s + 3·25-s + 12·29-s − 16·43-s − 4·49-s + 24·53-s − 16·67-s + 28·73-s + 16·95-s − 20·97-s + 12·101-s + 24·115-s + 20·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.83·19-s + 2.50·23-s + 3/5·25-s + 2.22·29-s − 2.43·43-s − 4/7·49-s + 3.29·53-s − 1.95·67-s + 3.27·73-s + 1.64·95-s − 2.03·97-s + 1.19·101-s + 2.23·115-s + 1.81·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.569412823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569412823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.583201085652158121363662223132, −9.552877189111995015695588093220, −8.755516426329852982150746987352, −8.707388331462401126209003149596, −8.302460099068568076100880909309, −7.63279126325622131410266809315, −7.10325057521550804251878034891, −7.04954946287071256074005279456, −6.37704715052509634754946576948, −6.22965967969786156488872182174, −5.31927896366760080961177852302, −5.17065473681717308899549584736, −5.00412621955729313779094173007, −4.30172953248277263236832492574, −3.55333840883406184357351111498, −3.07371350998390021116684778835, −2.79096812173766810479584479102, −2.10397117970830821193183056308, −1.19700913620606236227303931417, −0.926426825915562155868983867744,
0.926426825915562155868983867744, 1.19700913620606236227303931417, 2.10397117970830821193183056308, 2.79096812173766810479584479102, 3.07371350998390021116684778835, 3.55333840883406184357351111498, 4.30172953248277263236832492574, 5.00412621955729313779094173007, 5.17065473681717308899549584736, 5.31927896366760080961177852302, 6.22965967969786156488872182174, 6.37704715052509634754946576948, 7.04954946287071256074005279456, 7.10325057521550804251878034891, 7.63279126325622131410266809315, 8.302460099068568076100880909309, 8.707388331462401126209003149596, 8.755516426329852982150746987352, 9.552877189111995015695588093220, 9.583201085652158121363662223132
