| 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
| \(L(1)\) |
\(\approx\) |
\(0.6606846268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6606846268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.814664858414589688705655542096, −9.545668919304885953895126202993, −8.960821570036293786915612412004, −8.259802417940044400654304209173, −8.164200710749238443093188593129, −7.63099950730960849041949881857, −7.60089067911024098961765696432, −6.91303808038793915420660093173, −6.51102907695182650036893253737, −6.03956736963881357444584834575, −5.58668786992475249138219187811, −5.09465913109846719734902696805, −4.76573620317319398993794520093, −4.05845767456516893115500909095, −3.72389960287801075244983430054, −3.29978482382380789423498193042, −2.85963911508421866259496146784, −1.71515048087369980231577914394, −1.70792831392382837476634408737, −0.32018988291142652546488151022,
0.32018988291142652546488151022, 1.70792831392382837476634408737, 1.71515048087369980231577914394, 2.85963911508421866259496146784, 3.29978482382380789423498193042, 3.72389960287801075244983430054, 4.05845767456516893115500909095, 4.76573620317319398993794520093, 5.09465913109846719734902696805, 5.58668786992475249138219187811, 6.03956736963881357444584834575, 6.51102907695182650036893253737, 6.91303808038793915420660093173, 7.60089067911024098961765696432, 7.63099950730960849041949881857, 8.164200710749238443093188593129, 8.259802417940044400654304209173, 8.960821570036293786915612412004, 9.545668919304885953895126202993, 9.814664858414589688705655542096
