| L(s) = 1 | − 2·4-s − 4·7-s + 3·16-s − 8·25-s + 8·28-s − 8·37-s + 16·43-s − 2·49-s − 4·64-s + 32·67-s − 40·79-s − 9·81-s + 16·100-s + 40·109-s − 12·112-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 3/4·16-s − 8/5·25-s + 1.51·28-s − 1.31·37-s + 2.43·43-s − 2/7·49-s − 1/2·64-s + 3.90·67-s − 4.50·79-s − 81-s + 8/5·100-s + 3.83·109-s − 1.13·112-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3010959365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3010959365\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38419946000972642730684544382, −11.85602180202949993513351099253, −11.31622493666485858414980334599, −11.13542303485531380590793216534, −11.04663477172047608634985054464, −10.06245360777631225887165823302, −10.01684414584459712677289350419, −9.983427976060114342601612940175, −9.619505688640539452442220136829, −9.188256232215632503344691705404, −8.716322088401785399914411583104, −8.572048130785580253217339229931, −8.257988799522835273490398596777, −7.51282438291152096907083617904, −7.40391623681593119943416594016, −6.96245898773232803232375345408, −6.45884431019503074904284879122, −5.85932993348368137884727578774, −5.84452205332241463593167315187, −5.30619479484682825402135209402, −4.50847591333173212551836116947, −4.30648197427357298034382340907, −3.46085368216767048570007564189, −3.38201100167339061682663398425, −2.27924580663826041386472562064,
2.27924580663826041386472562064, 3.38201100167339061682663398425, 3.46085368216767048570007564189, 4.30648197427357298034382340907, 4.50847591333173212551836116947, 5.30619479484682825402135209402, 5.84452205332241463593167315187, 5.85932993348368137884727578774, 6.45884431019503074904284879122, 6.96245898773232803232375345408, 7.40391623681593119943416594016, 7.51282438291152096907083617904, 8.257988799522835273490398596777, 8.572048130785580253217339229931, 8.716322088401785399914411583104, 9.188256232215632503344691705404, 9.619505688640539452442220136829, 9.983427976060114342601612940175, 10.01684414584459712677289350419, 10.06245360777631225887165823302, 11.04663477172047608634985054464, 11.13542303485531380590793216534, 11.31622493666485858414980334599, 11.85602180202949993513351099253, 12.38419946000972642730684544382
