| L(s) = 1 | − 4·4-s + 8·7-s + 10·16-s − 24·25-s − 32·28-s + 40·31-s + 52·49-s − 24·64-s + 24·73-s − 24·79-s − 32·97-s + 96·100-s + 72·103-s + 80·112-s − 24·121-s − 160·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 40·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3.02·7-s + 5/2·16-s − 4.79·25-s − 6.04·28-s + 7.18·31-s + 52/7·49-s − 3·64-s + 2.80·73-s − 2.70·79-s − 3.24·97-s + 48/5·100-s + 7.09·103-s + 7.55·112-s − 2.18·121-s − 14.3·124-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 − 3.07·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.29286846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.29286846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T^{2} + 3 p T^{4} + p^{3} T^{6} + 5 p^{2} T^{8} + p^{5} T^{10} + 3 p^{5} T^{12} + p^{8} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| good | 5 | \( ( 1 + 12 T^{2} + 66 T^{4} + 336 T^{6} + 1859 T^{8} + 336 p^{2} T^{10} + 66 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 12 T^{2} - 6 T^{4} - 1104 T^{6} - 9565 T^{8} - 1104 p^{2} T^{10} - 6 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 10 T^{2} + 235 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 12 T^{2} + 178 T^{4} + 432 p T^{6} - 429 p^{2} T^{8} + 432 p^{3} T^{10} + 178 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 66 T^{2} + 2553 T^{4} + 71346 T^{6} + 1547972 T^{8} + 71346 p^{2} T^{10} + 2553 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 68 T^{2} + 2418 T^{4} - 78064 T^{6} + 2140499 T^{8} - 78064 p^{2} T^{10} + 2418 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 52 T^{2} + 1846 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 10 T + 15 T^{2} - 230 T^{3} + 3164 T^{4} - 230 p T^{5} + 15 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + 6 T^{2} - 2583 T^{4} - 714 T^{6} + 4935716 T^{8} - 714 p^{2} T^{10} - 2583 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 68 T^{2} + 4390 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 50 T^{2} + 1435 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 28 T^{2} - 3182 T^{4} - 12656 T^{6} + 9117619 T^{8} - 12656 p^{2} T^{10} - 3182 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 108 T^{2} + 5442 T^{4} + 65232 T^{6} - 1941373 T^{8} + 65232 p^{2} T^{10} + 5442 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 100 T^{2} + 930 T^{4} + 210800 T^{6} + 35337539 T^{8} + 210800 p^{2} T^{10} + 930 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 164 T^{2} + 13242 T^{4} + 1018768 T^{6} + 72505859 T^{8} + 1018768 p^{2} T^{10} + 13242 p^{4} T^{12} + 164 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 98 T^{2} + 537 T^{4} + 8722 T^{6} + 18947012 T^{8} + 8722 p^{2} T^{10} + 537 p^{4} T^{12} + 98 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 36 T^{2} + 1694 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 + 6 T - 113 T^{2} - 54 T^{3} + 13116 T^{4} - 54 p T^{5} - 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 132 T^{2} + 15822 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 260 T^{2} + 34986 T^{4} - 4360720 T^{6} + 465471155 T^{8} - 4360720 p^{2} T^{10} + 34986 p^{4} T^{12} - 260 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
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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
−2.97134739031215570882176462964, −2.92989920035163267488480749590, −2.71591614145100259590656979226, −2.59855762953309073095810165074, −2.53441707303190712837617626732, −2.53020456016315852135223231442, −2.45123623179100946981161479966, −2.36536532969004961651361066328, −2.30848346336240223024397189531, −2.18973414188269638262773665181, −2.18003244141945899104708451941, −2.14328762822979588925389050060, −1.63835002451740464465185633333, −1.61494833828793549270432224130, −1.58160573580533612833718393668, −1.55743336657868097264563546475, −1.44444771847364729361040035116, −1.38283679065753218393440193246, −1.32629612755058273418819048084, −1.12261681376626241489783070779, −0.795193486975663015833426371400, −0.68366179958327286120407137546, −0.58348729836148805383237320781, −0.58170944483145305298450949053, −0.29600818430014018537028518613,
0.29600818430014018537028518613, 0.58170944483145305298450949053, 0.58348729836148805383237320781, 0.68366179958327286120407137546, 0.795193486975663015833426371400, 1.12261681376626241489783070779, 1.32629612755058273418819048084, 1.38283679065753218393440193246, 1.44444771847364729361040035116, 1.55743336657868097264563546475, 1.58160573580533612833718393668, 1.61494833828793549270432224130, 1.63835002451740464465185633333, 2.14328762822979588925389050060, 2.18003244141945899104708451941, 2.18973414188269638262773665181, 2.30848346336240223024397189531, 2.36536532969004961651361066328, 2.45123623179100946981161479966, 2.53020456016315852135223231442, 2.53441707303190712837617626732, 2.59855762953309073095810165074, 2.71591614145100259590656979226, 2.92989920035163267488480749590, 2.97134739031215570882176462964
Plot not available for L-functions of degree greater than 10.
