| L(s) = 1 | − 8·5-s + 36·25-s + 16·29-s + 16·49-s + 32·53-s − 16·73-s − 16·97-s − 48·101-s + 48·121-s − 120·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 96·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 36/5·25-s + 2.97·29-s + 16/7·49-s + 4.39·53-s − 1.87·73-s − 1.62·97-s − 4.77·101-s + 4.36·121-s − 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05335083709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05335083709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{8} \) |
| good | 7 | \( ( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 290 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 48 T^{2} + 2930 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 64 T^{2} + 4002 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 76 T^{2} + 4326 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 120 T^{2} + 9698 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T^{2} - 6378 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 140 T^{2} + 11238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 252 T^{2} + 28118 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 160 T^{2} + 12642 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.85348652003885104788723247860, −3.42777407623878629226922309844, −3.32576053661246620648815673544, −3.20167499063318925265639684952, −3.08575268387704176394524832059, −3.06085260184395142438531388991, −3.05287830881712473773621152978, −3.03049222002099720804535646465, −2.80181591545419151095093248274, −2.57480829485915101400480114897, −2.52105152667282608934595852246, −2.29146832634419190759476662080, −2.28243302151656026664887301743, −1.97895107213756471505625435332, −1.94435300811659237676581986362, −1.71118062415926661873327466079, −1.65808705228749387618822908173, −1.35418409779490291045070037717, −1.05230050884113857794439136339, −0.982877231401144464920100487816, −0.926627928347291665972406141261, −0.60919932718056149615885487039, −0.58890103957505284695471947573, −0.56854732198946360135114567244, −0.02258085243680182030232890715,
0.02258085243680182030232890715, 0.56854732198946360135114567244, 0.58890103957505284695471947573, 0.60919932718056149615885487039, 0.926627928347291665972406141261, 0.982877231401144464920100487816, 1.05230050884113857794439136339, 1.35418409779490291045070037717, 1.65808705228749387618822908173, 1.71118062415926661873327466079, 1.94435300811659237676581986362, 1.97895107213756471505625435332, 2.28243302151656026664887301743, 2.29146832634419190759476662080, 2.52105152667282608934595852246, 2.57480829485915101400480114897, 2.80181591545419151095093248274, 3.03049222002099720804535646465, 3.05287830881712473773621152978, 3.06085260184395142438531388991, 3.08575268387704176394524832059, 3.20167499063318925265639684952, 3.32576053661246620648815673544, 3.42777407623878629226922309844, 3.85348652003885104788723247860
Plot not available for L-functions of degree greater than 10.
