| L(s) = 1 | − 4·9-s − 6·11-s − 30·29-s − 12·31-s + 26·41-s + 9·49-s + 16·59-s + 26·61-s + 18·71-s + 42·79-s + 10·81-s − 24·89-s + 24·99-s + 62·101-s − 12·109-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 1.80·11-s − 5.57·29-s − 2.15·31-s + 4.06·41-s + 9/7·49-s + 2.08·59-s + 3.32·61-s + 2.13·71-s + 4.72·79-s + 10/9·81-s − 2.54·89-s + 2.41·99-s + 6.16·101-s − 1.14·109-s − 1.18·121-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 + 3·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2340295597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2340295597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 9 T^{2} + 139 T^{4} - 711 T^{6} + 7704 T^{8} - 711 p^{2} T^{10} + 139 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 3 T + 20 T^{2} + 54 T^{3} + 243 T^{4} + 54 p T^{5} + 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 3 p T^{2} + 790 T^{4} - 10368 T^{6} + 128673 T^{8} - 10368 p^{2} T^{10} + 790 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 65 T^{2} + 1633 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 39 T^{2} - 60 T^{3} + 44 p T^{4} - 60 p T^{5} + 39 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 123 T^{2} + 7330 T^{4} - 279288 T^{6} + 7526469 T^{8} - 279288 p^{2} T^{10} + 7330 p^{4} T^{12} - 123 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 15 T + 164 T^{2} + 1170 T^{3} + 7281 T^{4} + 1170 p T^{5} + 164 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + 3 p T^{2} + 318 T^{3} + 3539 T^{4} + 318 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 47 T^{2} + 2670 T^{4} - 90832 T^{6} + 5094089 T^{8} - 90832 p^{2} T^{10} + 2670 p^{4} T^{12} - 47 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 13 T + 155 T^{2} - 1249 T^{3} + 8848 T^{4} - 1249 p T^{5} + 155 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 125 T^{2} + 7053 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 183 T^{2} + 11962 T^{4} - 248976 T^{6} - 1203891 T^{8} - 248976 p^{2} T^{10} + 11962 p^{4} T^{12} - 183 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 301 T^{2} + 40599 T^{4} - 3376979 T^{6} + 203717984 T^{8} - 3376979 p^{2} T^{10} + 40599 p^{4} T^{12} - 301 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 8 T + 47 T^{2} + 574 T^{3} - 6221 T^{4} + 574 p T^{5} + 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 13 T + 121 T^{2} - 703 T^{3} + 3124 T^{4} - 703 p T^{5} + 121 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 3 T^{2} + 6415 T^{4} + 99 p T^{6} + 48836664 T^{8} + 99 p^{3} T^{10} + 6415 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 9 T + 283 T^{2} - 1797 T^{3} + 30024 T^{4} - 1797 p T^{5} + 283 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 377 T^{2} + 72387 T^{4} - 8942539 T^{6} + 772942784 T^{8} - 8942539 p^{2} T^{10} + 72387 p^{4} T^{12} - 377 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 21 T + 319 T^{2} - 3879 T^{3} + 37884 T^{4} - 3879 p T^{5} + 319 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 229 T^{2} + 37515 T^{4} - 49201 p T^{6} + 385185608 T^{8} - 49201 p^{3} T^{10} + 37515 p^{4} T^{12} - 229 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 12 T + 353 T^{2} + 3150 T^{3} + 46956 T^{4} + 3150 p T^{5} + 353 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 625 T^{2} + 181591 T^{4} - 32088775 T^{6} + 3771831376 T^{8} - 32088775 p^{2} T^{10} + 181591 p^{4} T^{12} - 625 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.38861659025515183525242837786, −3.37653686068264132714555389966, −2.94893224289645595460302696307, −2.94496304969601059052646909845, −2.77643375271936565162704064517, −2.73116255750232733030998399361, −2.53255577114478525743253537281, −2.38742402314291200695983636424, −2.37461751427821487030254901585, −2.23202021655212948037062778437, −2.19193377324302399889635487629, −2.15152711860714224420826447826, −1.99403846333524756531823879467, −1.92954107625152772597088603176, −1.90832263229045716149693332363, −1.50577344870171157545923384560, −1.43692979273384505555012382582, −1.15439249352552819897464043570, −1.05984166767295396924635449964, −0.927146285051308016546901074780, −0.852858671187509984375719036122, −0.53931515961035545194060220009, −0.39729088829953576766156842326, −0.37256297523656360201148427452, −0.04250184961434522257182240598,
0.04250184961434522257182240598, 0.37256297523656360201148427452, 0.39729088829953576766156842326, 0.53931515961035545194060220009, 0.852858671187509984375719036122, 0.927146285051308016546901074780, 1.05984166767295396924635449964, 1.15439249352552819897464043570, 1.43692979273384505555012382582, 1.50577344870171157545923384560, 1.90832263229045716149693332363, 1.92954107625152772597088603176, 1.99403846333524756531823879467, 2.15152711860714224420826447826, 2.19193377324302399889635487629, 2.23202021655212948037062778437, 2.37461751427821487030254901585, 2.38742402314291200695983636424, 2.53255577114478525743253537281, 2.73116255750232733030998399361, 2.77643375271936565162704064517, 2.94496304969601059052646909845, 2.94893224289645595460302696307, 3.37653686068264132714555389966, 3.38861659025515183525242837786
Plot not available for L-functions of degree greater than 10.
