| L(s) = 1 | + 2·3-s + 2·7-s + 3·9-s + 3·11-s + 13-s + 7·17-s + 6·19-s + 4·21-s + 4·27-s + 8·29-s − 31-s + 6·33-s + 3·37-s + 2·39-s − 11·41-s − 8·43-s + 2·47-s + 3·49-s + 14·51-s − 5·53-s + 12·57-s + 7·59-s − 9·61-s + 6·63-s − 5·67-s + 5·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s + 1.37·19-s + 0.872·21-s + 0.769·27-s + 1.48·29-s − 0.179·31-s + 1.04·33-s + 0.493·37-s + 0.320·39-s − 1.71·41-s − 1.21·43-s + 0.291·47-s + 3/7·49-s + 1.96·51-s − 0.686·53-s + 1.58·57-s + 0.911·59-s − 1.15·61-s + 0.755·63-s − 0.610·67-s + 0.593·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.000372715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.000372715\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 92 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 138 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 264 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 140 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.342891979245086082225955556313, −8.285884209370000934533375694845, −7.933099607192562504165377436001, −7.69433885345837310586538888996, −7.04549668953965699247789720128, −7.03320068261975204395455623540, −6.40542899448805416256246386319, −6.11880122393368794401802780325, −5.57274725721266400897769463590, −5.08042057591114403961014270961, −4.87620393910601210683448169534, −4.49811724753211738935695271051, −3.72182229444244686708406961900, −3.64846973434820364175435893626, −3.06837157215074315743802285666, −3.01144146190462939960755721361, −1.97748224428306969702760413757, −1.89177776843764691344070781077, −1.01793179492249247240349892114, −0.960338179719121518290184192895,
0.960338179719121518290184192895, 1.01793179492249247240349892114, 1.89177776843764691344070781077, 1.97748224428306969702760413757, 3.01144146190462939960755721361, 3.06837157215074315743802285666, 3.64846973434820364175435893626, 3.72182229444244686708406961900, 4.49811724753211738935695271051, 4.87620393910601210683448169534, 5.08042057591114403961014270961, 5.57274725721266400897769463590, 6.11880122393368794401802780325, 6.40542899448805416256246386319, 7.03320068261975204395455623540, 7.04549668953965699247789720128, 7.69433885345837310586538888996, 7.933099607192562504165377436001, 8.285884209370000934533375694845, 8.342891979245086082225955556313
