| L(s) = 1 | + 4·3-s − 2·9-s − 40·27-s − 36·41-s − 16·43-s + 4·49-s − 44·67-s − 55·81-s + 60·83-s + 12·89-s + 36·107-s + 26·121-s − 144·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2/3·9-s − 7.69·27-s − 5.62·41-s − 2.43·43-s + 4/7·49-s − 5.37·67-s − 6.11·81-s + 6.58·83-s + 1.27·89-s + 3.48·107-s + 2.36·121-s − 12.9·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434896439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434896439\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.66808310543572713742591452838, −6.41351882844141578071343444662, −6.31790909148739667868737525924, −6.11317597043784557410466137198, −5.82350585779589843475681879449, −5.66659117529891332095836629974, −5.47527955356213289076509702999, −4.99176828358205324890775730627, −4.89976401236657384114025377227, −4.79148554203322882464100533833, −4.74866849081609263614755408604, −4.05535092209468176653886286985, −3.67330014444403732992126931491, −3.49680976537946342037117879445, −3.45487455866185553177947095519, −3.39795133486916988462435407747, −2.96174008167483268024436808646, −2.83579584084404623751180765951, −2.72011967522874503846856554811, −2.00818428692560639538080067072, −1.97888845704182359167537715346, −1.81290984275541524053247721743, −1.66144543216919699029578046431, −0.57711382498361898731068023806, −0.31158091186436398135864815359,
0.31158091186436398135864815359, 0.57711382498361898731068023806, 1.66144543216919699029578046431, 1.81290984275541524053247721743, 1.97888845704182359167537715346, 2.00818428692560639538080067072, 2.72011967522874503846856554811, 2.83579584084404623751180765951, 2.96174008167483268024436808646, 3.39795133486916988462435407747, 3.45487455866185553177947095519, 3.49680976537946342037117879445, 3.67330014444403732992126931491, 4.05535092209468176653886286985, 4.74866849081609263614755408604, 4.79148554203322882464100533833, 4.89976401236657384114025377227, 4.99176828358205324890775730627, 5.47527955356213289076509702999, 5.66659117529891332095836629974, 5.82350585779589843475681879449, 6.11317597043784557410466137198, 6.31790909148739667868737525924, 6.41351882844141578071343444662, 6.66808310543572713742591452838
