| L(s) = 1 | − 5·16-s − 12·25-s − 16·43-s + 16·67-s + 80·73-s − 64·97-s − 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 5/4·16-s − 2.39·25-s − 2.43·43-s + 1.95·67-s + 9.36·73-s − 6.49·97-s − 9.45·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4165994532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4165994532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5 T^{4} + 7 p^{2} T^{8} + 5 p^{4} T^{12} + p^{8} T^{16} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + 6 T^{2} + 18 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| good | 7 | \( ( 1 - 96 T^{4} + 5630 T^{8} - 96 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 26 T^{2} + 370 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 - 496 T^{4} + 110590 T^{8} - 496 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 176 T^{4} - 3810 T^{8} - 176 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{8} \) |
| 23 | \( ( 1 - 156 T^{4} - 284410 T^{8} - 156 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 2 p T^{2} + 2482 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 48 T^{2} + 2334 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + 656 T^{4} + 3359806 T^{8} + 656 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 72 T^{2} + 3182 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 4 T + 8 T^{2} - 148 T^{3} - 3662 T^{4} - 148 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 700 T^{4} - 5274362 T^{8} - 700 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 6144 T^{4} + 23799710 T^{8} + 6144 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 218 T^{2} + 18802 T^{4} - 218 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 24 T^{2} + 3486 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 4 T + 8 T^{2} + 52 T^{3} - 6062 T^{4} + 52 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - p T^{2} )^{16} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{8}( 1 + 6 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 240 T^{2} + 26718 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 19036 T^{4} + 182821990 T^{8} - 19036 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 8 T^{2} - 11858 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 752 T^{3} + 1918 T^{4} + 752 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.31826003668687123890945734712, −3.17864467275976402677148013726, −3.02004917380948074744113320179, −2.88908348318522249619257616733, −2.73802737202922047599124084041, −2.62990210467620933289652238909, −2.60515337348381467003967154380, −2.53959667501647327406557116579, −2.44087295412223992968608576010, −2.40637245918648033406026770362, −2.38061731847650146899311985544, −2.15031329907333678022553474693, −2.07416664018223191149586752754, −1.97728256169692771069044397300, −1.75249608383752038717588724173, −1.72098773052178247465044446417, −1.62171367303887399548117714078, −1.62080084818696585843394861104, −1.47961314524814336761251095546, −0.996036944757701460424364819059, −0.979278526804446615595442368149, −0.941359490234106043160903819761, −0.937107379555204188875558241873, −0.23643089079181000691966156198, −0.14312749591513282974425955231,
0.14312749591513282974425955231, 0.23643089079181000691966156198, 0.937107379555204188875558241873, 0.941359490234106043160903819761, 0.979278526804446615595442368149, 0.996036944757701460424364819059, 1.47961314524814336761251095546, 1.62080084818696585843394861104, 1.62171367303887399548117714078, 1.72098773052178247465044446417, 1.75249608383752038717588724173, 1.97728256169692771069044397300, 2.07416664018223191149586752754, 2.15031329907333678022553474693, 2.38061731847650146899311985544, 2.40637245918648033406026770362, 2.44087295412223992968608576010, 2.53959667501647327406557116579, 2.60515337348381467003967154380, 2.62990210467620933289652238909, 2.73802737202922047599124084041, 2.88908348318522249619257616733, 3.02004917380948074744113320179, 3.17864467275976402677148013726, 3.31826003668687123890945734712
Plot not available for L-functions of degree greater than 10.
