| L(s) = 1 | + 4·3-s + 4·9-s − 4·27-s − 24·31-s + 24·37-s + 24·41-s + 20·43-s + 4·49-s − 20·67-s + 24·71-s + 48·79-s − 10·81-s + 12·83-s + 24·89-s − 96·93-s + 60·107-s + 96·111-s + 20·121-s + 96·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4/3·9-s − 0.769·27-s − 4.31·31-s + 3.94·37-s + 3.74·41-s + 3.04·43-s + 4/7·49-s − 2.44·67-s + 2.84·71-s + 5.40·79-s − 1.11·81-s + 1.31·83-s + 2.54·89-s − 9.95·93-s + 5.80·107-s + 9.11·111-s + 1.81·121-s + 8.65·123-s + 0.0887·127-s + 7.04·129-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.46925312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.46925312\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.65948532230546944241151172917, −6.45903669011456216876794986536, −6.20429962429969083927797120553, −6.11056806771401984182229514169, −5.86448894286268259036193452799, −5.59332107084300178945845068308, −5.51193255205727252338744014877, −5.34466375302736938151151565560, −4.72489100613936883255641546037, −4.55365316953575248637947971889, −4.52205395001125350466114157894, −4.21221432115153884411414181234, −3.93370253131450462495579652044, −3.59918204684581914792743069194, −3.31909520095100338431866862732, −3.30697522565437927033711087196, −3.27641404066480984519586420917, −2.46102723005948952091410348880, −2.44557022853871952542399460665, −2.24132143991853128225048977259, −2.21168102323827038235937310569, −1.98417071091467643830613287298, −1.03133021652598914611604034506, −0.847790244608192170227646061092, −0.63154940203184196003722781810,
0.63154940203184196003722781810, 0.847790244608192170227646061092, 1.03133021652598914611604034506, 1.98417071091467643830613287298, 2.21168102323827038235937310569, 2.24132143991853128225048977259, 2.44557022853871952542399460665, 2.46102723005948952091410348880, 3.27641404066480984519586420917, 3.30697522565437927033711087196, 3.31909520095100338431866862732, 3.59918204684581914792743069194, 3.93370253131450462495579652044, 4.21221432115153884411414181234, 4.52205395001125350466114157894, 4.55365316953575248637947971889, 4.72489100613936883255641546037, 5.34466375302736938151151565560, 5.51193255205727252338744014877, 5.59332107084300178945845068308, 5.86448894286268259036193452799, 6.11056806771401984182229514169, 6.20429962429969083927797120553, 6.45903669011456216876794986536, 6.65948532230546944241151172917
