| L(s) = 1 | − 5·7-s − 12·13-s + 5·25-s + 18·31-s + 20·37-s + 8·43-s + 18·49-s − 12·61-s + 16·67-s + 4·79-s + 60·91-s − 24·97-s − 6·103-s − 4·109-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 83·169-s + 173-s − 25·175-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 3.32·13-s + 25-s + 3.23·31-s + 3.28·37-s + 1.21·43-s + 18/7·49-s − 1.53·61-s + 1.95·67-s + 0.450·79-s + 6.28·91-s − 2.43·97-s − 0.591·103-s − 0.383·109-s − 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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.38·169-s + 0.0760·173-s − 1.88·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287124176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287124176\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.387809508975689000249585197219, −9.103509205428981506949986149378, −8.208589408326053278666521670096, −8.154327080945006908987647372339, −7.60554665550274450709769584238, −7.22999512467643046274011149072, −6.83809692559673156151735284409, −6.64093175316774319139620597647, −6.04739031360627796920337359667, −5.88768310434426072934932703449, −5.16195641492979324546660619485, −4.78726838315492925251672094354, −4.32758601381555243572485351093, −4.19610009894454051298856296203, −3.18981407146357412907251016673, −2.81898424462542065579977145911, −2.58412056679973464806434388687, −2.32476391743353586519412957773, −0.944152454577388731468639821574, −0.48208496981405479155920006127,
0.48208496981405479155920006127, 0.944152454577388731468639821574, 2.32476391743353586519412957773, 2.58412056679973464806434388687, 2.81898424462542065579977145911, 3.18981407146357412907251016673, 4.19610009894454051298856296203, 4.32758601381555243572485351093, 4.78726838315492925251672094354, 5.16195641492979324546660619485, 5.88768310434426072934932703449, 6.04739031360627796920337359667, 6.64093175316774319139620597647, 6.83809692559673156151735284409, 7.22999512467643046274011149072, 7.60554665550274450709769584238, 8.154327080945006908987647372339, 8.208589408326053278666521670096, 9.103509205428981506949986149378, 9.387809508975689000249585197219
