| L(s) = 1 | + 6·5-s − 3·9-s + 10·13-s + 60·17-s + 61·25-s + 66·29-s − 40·37-s − 144·41-s − 18·45-s − 97·49-s − 360·53-s + 14·61-s + 60·65-s − 220·73-s + 81·81-s + 360·85-s + 912·89-s + 200·97-s + 198·101-s − 248·109-s − 570·113-s − 30·117-s − 214·121-s + 270·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 1/3·9-s + 0.769·13-s + 3.52·17-s + 2.43·25-s + 2.27·29-s − 1.08·37-s − 3.51·41-s − 2/5·45-s − 1.97·49-s − 6.79·53-s + 0.229·61-s + 0.923·65-s − 3.01·73-s + 81-s + 4.23·85-s + 10.2·89-s + 2.06·97-s + 1.96·101-s − 2.27·109-s − 5.04·113-s − 0.256·117-s − 1.76·121-s + 2.15·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.159939222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.159939222\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T^{2} - 8 p^{2} T^{4} + p^{5} T^{6} + p^{8} T^{8} \) |
| good | 5 | \( ( 1 - 3 T - 17 T^{2} + 72 T^{3} - 174 T^{4} + 72 p^{2} T^{5} - 17 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 + 97 T^{2} + 4381 T^{4} + 21922 T^{6} - 3075026 T^{8} + 21922 p^{4} T^{10} + 4381 p^{8} T^{12} + 97 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 107 T^{2} - 3192 T^{4} + 107 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5 T - 83 T^{2} + 1150 T^{3} - 21122 T^{4} + 1150 p^{2} T^{5} - 83 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 15 T + 608 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 31 p T^{2} + 294216 T^{4} - 31 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 503 T^{2} - 197699 T^{4} + 103618 p^{2} T^{6} + 193054 p^{4} T^{8} + 103618 p^{6} T^{10} - 197699 p^{8} T^{12} - 503 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 33 T - 839 T^{2} - 8118 T^{3} + 2094054 T^{4} - 8118 p^{2} T^{5} - 839 p^{4} T^{6} - 33 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 251 T^{2} + 804865 T^{4} + 649815406 T^{6} - 360711589466 T^{8} + 649815406 p^{4} T^{10} + 804865 p^{8} T^{12} - 251 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 10 T + 1818 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 72 T + 1471 T^{2} + 25272 T^{3} + 2246304 T^{4} + 25272 p^{2} T^{5} + 1471 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 + 3166 T^{2} + 4082065 T^{4} - 2837087426 T^{6} - 10655428250876 T^{8} - 2837087426 p^{4} T^{10} + 4082065 p^{8} T^{12} + 3166 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 + 2977 T^{2} + 5326621 T^{4} - 18527222558 T^{6} - 54554955650546 T^{8} - 18527222558 p^{4} T^{10} + 5326621 p^{8} T^{12} + 2977 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 90 T + 6698 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 3587 T^{2} + 749208 T^{4} + 3587 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7 T - 7169 T^{2} + 1568 T^{3} + 38071354 T^{4} + 1568 p^{2} T^{5} - 7169 p^{4} T^{6} - 7 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8411 T^{2} + 50593800 T^{4} + 8411 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 7922 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 55 T + 5508 T^{2} + 55 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( 1 + 20869 T^{2} + 251340865 T^{4} + 2217834902446 T^{6} + 15439711699511014 T^{8} + 2217834902446 p^{4} T^{10} + 251340865 p^{8} T^{12} + 20869 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 + 16837 T^{2} + 138536161 T^{4} + 842384844142 T^{6} + 5304047075446774 T^{8} + 842384844142 p^{4} T^{10} + 138536161 p^{8} T^{12} + 16837 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 228 T + 27158 T^{2} - 228 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 100 T - 29 p T^{2} + 600500 T^{3} - 18446312 T^{4} + 600500 p^{2} T^{5} - 29 p^{5} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.69618342072790520523775242123, −4.33087399157782958090700234332, −4.24128204358441777189135893472, −3.82650760221494073796567852328, −3.75950275445849861888823960955, −3.73528423406373206478794821898, −3.39448384832021430054661154326, −3.39394029256216867932376974227, −3.34766787798366273153207278665, −3.03160754344671054405877375854, −2.99548646711911654504632740385, −2.92762050473345359294965228569, −2.90662138518172555255369149993, −2.55949665659481282136843862279, −2.33402709535997750041306281741, −1.89317875116005807434755358156, −1.82345522712763143334757771238, −1.73069692425097924176630985616, −1.69002325943919251853949972081, −1.36138890999409085298049751888, −1.17195720124470106972130620136, −1.11930156279227432772921768922, −0.62596296210864487254388992637, −0.60547067117465502806125224357, −0.17165631004554358879165385442,
0.17165631004554358879165385442, 0.60547067117465502806125224357, 0.62596296210864487254388992637, 1.11930156279227432772921768922, 1.17195720124470106972130620136, 1.36138890999409085298049751888, 1.69002325943919251853949972081, 1.73069692425097924176630985616, 1.82345522712763143334757771238, 1.89317875116005807434755358156, 2.33402709535997750041306281741, 2.55949665659481282136843862279, 2.90662138518172555255369149993, 2.92762050473345359294965228569, 2.99548646711911654504632740385, 3.03160754344671054405877375854, 3.34766787798366273153207278665, 3.39394029256216867932376974227, 3.39448384832021430054661154326, 3.73528423406373206478794821898, 3.75950275445849861888823960955, 3.82650760221494073796567852328, 4.24128204358441777189135893472, 4.33087399157782958090700234332, 4.69618342072790520523775242123
Plot not available for L-functions of degree greater than 10.
