| L(s) = 1 | − 4-s + 16-s + 10·17-s + 12·23-s + 9·25-s + 18·29-s − 20·43-s + 10·49-s + 2·53-s − 22·61-s − 64-s − 10·68-s − 8·79-s − 12·92-s − 9·100-s − 10·101-s − 20·103-s + 36·107-s + 6·113-s − 18·116-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s + 2.42·17-s + 2.50·23-s + 9/5·25-s + 3.34·29-s − 3.04·43-s + 10/7·49-s + 0.274·53-s − 2.81·61-s − 1/8·64-s − 1.21·68-s − 0.900·79-s − 1.25·92-s − 0.899·100-s − 0.995·101-s − 1.97·103-s + 3.48·107-s + 0.564·113-s − 1.67·116-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.301166496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.301166496\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.711970542738222295070650910758, −8.610955539379960473897136889740, −8.322755987115494943467151188484, −7.83907436373047634081940558220, −7.29463766894564383949426720923, −7.05424335617808908960899032481, −6.80351661624900700002831139825, −6.09152821689985624184002183231, −6.03046640692551337699466086062, −5.15315050556572388238775114744, −5.13802075929911804206259877187, −4.62117620225369492901933203560, −4.55024058224659873234591280012, −3.49281617707205192314307560699, −3.30330089232670758281480295660, −2.94261194782593428321256119274, −2.63412177212870195304451415281, −1.45004895937092057792652894105, −1.15426590914860982866423572037, −0.70795552803795409786794764343,
0.70795552803795409786794764343, 1.15426590914860982866423572037, 1.45004895937092057792652894105, 2.63412177212870195304451415281, 2.94261194782593428321256119274, 3.30330089232670758281480295660, 3.49281617707205192314307560699, 4.55024058224659873234591280012, 4.62117620225369492901933203560, 5.13802075929911804206259877187, 5.15315050556572388238775114744, 6.03046640692551337699466086062, 6.09152821689985624184002183231, 6.80351661624900700002831139825, 7.05424335617808908960899032481, 7.29463766894564383949426720923, 7.83907436373047634081940558220, 8.322755987115494943467151188484, 8.610955539379960473897136889740, 8.711970542738222295070650910758
