| L(s) = 1 | − 64·4-s − 324·9-s − 804·11-s + 2.56e3·16-s + 3.32e3·19-s + 1.14e4·29-s − 2.60e4·31-s + 2.07e4·36-s + 2.63e4·41-s + 5.14e4·44-s + 7.52e4·49-s + 2.16e4·59-s − 9.73e4·61-s − 8.19e4·64-s − 1.38e5·71-s − 2.12e5·76-s + 3.46e5·79-s + 6.56e4·81-s − 1.23e5·89-s + 2.60e5·99-s − 3.04e5·101-s + 5.92e5·109-s − 7.29e5·116-s − 6.53e5·121-s + 1.66e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4/3·9-s − 2.00·11-s + 5/2·16-s + 2.10·19-s + 2.51·29-s − 4.85·31-s + 8/3·36-s + 2.44·41-s + 4.00·44-s + 4.47·49-s + 0.810·59-s − 3.34·61-s − 5/2·64-s − 3.25·71-s − 4.21·76-s + 6.24·79-s + 10/9·81-s − 1.65·89-s + 2.67·99-s − 2.96·101-s + 4.77·109-s − 5.03·116-s − 4.05·121-s + 9.71·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8164885104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8164885104\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 75260 T^{2} + 3125887956 T^{4} - 85115406738820 T^{6} + 1677709563720878486 T^{8} - 85115406738820 p^{10} T^{10} + 3125887956 p^{20} T^{12} - 75260 p^{30} T^{14} + p^{40} T^{16} \) |
| 11 | \( ( 1 + 402 T + 569128 T^{2} + 151579554 T^{3} + 127773823230 T^{4} + 151579554 p^{5} T^{5} + 569128 p^{10} T^{6} + 402 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 13 | \( 1 - 1149060 T^{2} + 767812743556 T^{4} - 341624738567451420 T^{6} + \)\(13\!\cdots\!86\)\( T^{8} - 341624738567451420 p^{10} T^{10} + 767812743556 p^{20} T^{12} - 1149060 p^{30} T^{14} + p^{40} T^{16} \) |
| 17 | \( 1 - 5361250 T^{2} + 17665900265811 T^{4} - 38909069306572952900 T^{6} + \)\(64\!\cdots\!01\)\( T^{8} - 38909069306572952900 p^{10} T^{10} + 17665900265811 p^{20} T^{12} - 5361250 p^{30} T^{14} + p^{40} T^{16} \) |
| 19 | \( ( 1 - 1660 T + 2513511 T^{2} - 3343949870 T^{3} + 10288337090351 T^{4} - 3343949870 p^{5} T^{5} + 2513511 p^{10} T^{6} - 1660 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 23 | \( 1 - 21575190 T^{2} + 222347300835031 T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} - \)\(16\!\cdots\!80\)\( p^{10} T^{10} + 222347300835031 p^{20} T^{12} - 21575190 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( ( 1 - 5700 T + 67770536 T^{2} - 231367885500 T^{3} + 1796258986495726 T^{4} - 231367885500 p^{5} T^{5} + 67770536 p^{10} T^{6} - 5700 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 13002 T + 143542953 T^{2} + 1051915934454 T^{3} + 6516793832461955 T^{4} + 1051915934454 p^{5} T^{5} + 143542953 p^{10} T^{6} + 13002 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( 1 - 179258480 T^{2} + 21576263550619356 T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!86\)\( T^{8} - \)\(19\!\cdots\!60\)\( p^{10} T^{10} + 21576263550619356 p^{20} T^{12} - 179258480 p^{30} T^{14} + p^{40} T^{16} \) |
| 41 | \( ( 1 - 13158 T + 195986288 T^{2} + 3768584094 p T^{3} + 301897792030 p^{2} T^{4} + 3768584094 p^{6} T^{5} + 195986288 p^{10} T^{6} - 13158 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( 1 - 674816880 T^{2} + 236507642208199036 T^{4} - \)\(55\!\cdots\!60\)\( T^{6} + \)\(95\!\cdots\!26\)\( T^{8} - \)\(55\!\cdots\!60\)\( p^{10} T^{10} + 236507642208199036 p^{20} T^{12} - 674816880 p^{30} T^{14} + p^{40} T^{16} \) |
| 47 | \( 1 - 574274110 T^{2} + 194571789367212231 T^{4} - \)\(51\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} - \)\(51\!\cdots\!20\)\( p^{10} T^{10} + 194571789367212231 p^{20} T^{12} - 574274110 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( 1 - 2549076130 T^{2} + 3063127418185409011 T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} - \)\(22\!\cdots\!60\)\( p^{10} T^{10} + 3063127418185409011 p^{20} T^{12} - 2549076130 p^{30} T^{14} + p^{40} T^{16} \) |
| 59 | \( ( 1 - 10830 T + 1960546596 T^{2} - 5448340971510 T^{3} + 1706357341548599606 T^{4} - 5448340971510 p^{5} T^{5} + 1960546596 p^{10} T^{6} - 10830 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 + 48662 T + 3154387743 T^{2} + 102684066990304 T^{3} + 4008252461596535105 T^{4} + 102684066990304 p^{5} T^{5} + 3154387743 p^{10} T^{6} + 48662 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( 1 - 3928232180 T^{2} + 8112465026126458356 T^{4} - \)\(11\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!86\)\( T^{8} - \)\(11\!\cdots\!60\)\( p^{10} T^{10} + 8112465026126458356 p^{20} T^{12} - 3928232180 p^{30} T^{14} + p^{40} T^{16} \) |
| 71 | \( ( 1 + 69222 T + 8043922308 T^{2} + 374005340937654 T^{3} + 22513369265961569030 T^{4} + 374005340937654 p^{5} T^{5} + 8043922308 p^{10} T^{6} + 69222 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 - 12789390860 T^{2} + 77114344089266636356 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + \)\(71\!\cdots\!86\)\( T^{8} - \)\(28\!\cdots\!20\)\( p^{10} T^{10} + 77114344089266636356 p^{20} T^{12} - 12789390860 p^{30} T^{14} + p^{40} T^{16} \) |
| 79 | \( ( 1 - 173280 T + 16052618811 T^{2} - 1060162488622110 T^{3} + 61033830987150057551 T^{4} - 1060162488622110 p^{5} T^{5} + 16052618811 p^{10} T^{6} - 173280 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 1692464790 T^{2} + 49475711155407639031 T^{4} - \)\(75\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} - \)\(75\!\cdots\!80\)\( p^{10} T^{10} + 49475711155407639031 p^{20} T^{12} - 1692464790 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 + 61800 T + 7090434296 T^{2} + 439125085884600 T^{3} + 11322379633679008606 T^{4} + 439125085884600 p^{5} T^{5} + 7090434296 p^{10} T^{6} + 61800 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 97 | \( 1 - 22999744500 T^{2} + \)\(39\!\cdots\!56\)\( T^{4} - \)\(47\!\cdots\!00\)\( T^{6} + \)\(44\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!00\)\( p^{10} T^{10} + \)\(39\!\cdots\!56\)\( p^{20} T^{12} - 22999744500 p^{30} T^{14} + p^{40} T^{16} \) |
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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
−3.68594493986628230787912964746, −3.66349676987453568486751045532, −3.41175737517509180944700444184, −3.19845166687259826706977926093, −3.04308550966771693134668541099, −3.03338997331062738639782314447, −2.87357914633072568899640426042, −2.70744237576624168582106975379, −2.63484761747393491927851011030, −2.50618795783949398873776117377, −2.32695949222754941612468424143, −2.22601090120186945000036110134, −1.93518942368014676253736416679, −1.85288795130617038713940813308, −1.59492340843643358264391328624, −1.53955536489045029377544098926, −1.31217032226119584662627619885, −1.00898731511516196236038006084, −0.909979239166037053382155122329, −0.866226623823207205265085493300, −0.807977757307184502464449692421, −0.44935415916759333998778139445, −0.33551446271676391298902253303, −0.27967247296381290373244936650, −0.088508989762126420525060927165,
0.088508989762126420525060927165, 0.27967247296381290373244936650, 0.33551446271676391298902253303, 0.44935415916759333998778139445, 0.807977757307184502464449692421, 0.866226623823207205265085493300, 0.909979239166037053382155122329, 1.00898731511516196236038006084, 1.31217032226119584662627619885, 1.53955536489045029377544098926, 1.59492340843643358264391328624, 1.85288795130617038713940813308, 1.93518942368014676253736416679, 2.22601090120186945000036110134, 2.32695949222754941612468424143, 2.50618795783949398873776117377, 2.63484761747393491927851011030, 2.70744237576624168582106975379, 2.87357914633072568899640426042, 3.03338997331062738639782314447, 3.04308550966771693134668541099, 3.19845166687259826706977926093, 3.41175737517509180944700444184, 3.66349676987453568486751045532, 3.68594493986628230787912964746
Plot not available for L-functions of degree greater than 10.
