| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 2·7-s + 4·8-s + 9-s + 6·12-s + 4·14-s + 5·16-s + 2·18-s + 2·19-s + 4·21-s + 8·24-s + 8·25-s − 4·27-s + 6·28-s + 12·29-s + 6·32-s + 3·36-s + 4·38-s + 24·41-s + 8·42-s − 20·43-s + 10·48-s + 3·49-s + 16·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 1/3·9-s + 1.73·12-s + 1.06·14-s + 5/4·16-s + 0.471·18-s + 0.458·19-s + 0.872·21-s + 1.63·24-s + 8/5·25-s − 0.769·27-s + 1.13·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.648·38-s + 3.74·41-s + 1.23·42-s − 3.04·43-s + 1.44·48-s + 3/7·49-s + 2.26·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.710589250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.710589250\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.51754188692445679115250579128, −10.29894141647127017285802997633, −9.522875094748821951258836160293, −9.116965663169648458130078744279, −8.824622579405217990478505357295, −8.180517155663683647477521586017, −7.82260574882998311517653071107, −7.62210991110898862058755332820, −6.96789932483343091578826289210, −6.50637189096768429235471043318, −5.99424266465843508849610533626, −5.65245385407733710000746008718, −4.73284209205011835816238949216, −4.60023173225487087337730436500, −4.38483693080814909810985304316, −3.28179800241206653374697579559, −2.97780708836577916837175440405, −2.84027412297548881264682365945, −1.82277087557926880495454992444, −1.28960949145308200730985514293,
1.28960949145308200730985514293, 1.82277087557926880495454992444, 2.84027412297548881264682365945, 2.97780708836577916837175440405, 3.28179800241206653374697579559, 4.38483693080814909810985304316, 4.60023173225487087337730436500, 4.73284209205011835816238949216, 5.65245385407733710000746008718, 5.99424266465843508849610533626, 6.50637189096768429235471043318, 6.96789932483343091578826289210, 7.62210991110898862058755332820, 7.82260574882998311517653071107, 8.180517155663683647477521586017, 8.824622579405217990478505357295, 9.116965663169648458130078744279, 9.522875094748821951258836160293, 10.29894141647127017285802997633, 10.51754188692445679115250579128
