| L(s) = 1 | − 24·3-s + 10·4-s + 324·9-s − 240·12-s + 23·16-s − 196·17-s − 208·23-s + 558·25-s − 3.24e3·27-s + 388·29-s + 3.24e3·36-s − 900·43-s − 552·48-s + 302·49-s + 4.70e3·51-s + 524·53-s − 1.85e3·61-s − 120·64-s − 1.96e3·68-s + 4.99e3·69-s − 1.33e4·75-s − 1.49e3·79-s + 2.67e4·81-s − 9.31e3·87-s − 2.08e3·92-s + 5.58e3·100-s − 6.00e3·101-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 5/4·4-s + 12·9-s − 5.77·12-s + 0.359·16-s − 2.79·17-s − 1.88·23-s + 4.46·25-s − 23.0·27-s + 2.48·29-s + 15·36-s − 3.19·43-s − 1.65·48-s + 0.880·49-s + 12.9·51-s + 1.35·53-s − 3.89·61-s − 0.234·64-s − 3.49·68-s + 8.70·69-s − 20.6·75-s − 2.12·79-s + 36.6·81-s − 11.4·87-s − 2.35·92-s + 5.57·100-s − 5.91·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2319989698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2319989698\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T )^{8} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5 p T^{2} + 77 T^{4} - 105 p^{2} T^{6} + 1249 p^{2} T^{8} - 105 p^{8} T^{10} + 77 p^{12} T^{12} - 5 p^{19} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 - 558 T^{2} + 161249 T^{4} - 6182262 p T^{6} + 4417217556 T^{8} - 6182262 p^{7} T^{10} + 161249 p^{12} T^{12} - 558 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 - 302 T^{2} + 42335 p T^{4} - 109247814 T^{6} + 44429007764 T^{8} - 109247814 p^{6} T^{10} + 42335 p^{13} T^{12} - 302 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 6536 T^{2} + 21845948 T^{4} - 47897882232 T^{6} + 74648290193702 T^{8} - 47897882232 p^{6} T^{10} + 21845948 p^{12} T^{12} - 6536 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( ( 1 + 98 T + 12397 T^{2} + 986190 T^{3} + 96109736 T^{4} + 986190 p^{3} T^{5} + 12397 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 - 27632 T^{2} + 421589372 T^{4} - 4442768282256 T^{6} + 34687151257068134 T^{8} - 4442768282256 p^{6} T^{10} + 421589372 p^{12} T^{12} - 27632 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 + 104 T + 41844 T^{2} + 3340584 T^{3} + 719588614 T^{4} + 3340584 p^{3} T^{5} + 41844 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 194 T + 64761 T^{2} - 7214754 T^{3} + 1694674348 T^{4} - 7214754 p^{3} T^{5} + 64761 p^{6} T^{6} - 194 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( 1 - 2442 p T^{2} + 3396080265 T^{4} - 110316292290374 T^{6} + 3080244067782120564 T^{8} - 110316292290374 p^{6} T^{10} + 3396080265 p^{12} T^{12} - 2442 p^{19} T^{14} + p^{24} T^{16} \) |
| 37 | \( 1 - 376158 T^{2} + 63286170705 T^{4} - 6214893437597342 T^{6} + \)\(38\!\cdots\!96\)\( T^{8} - 6214893437597342 p^{6} T^{10} + 63286170705 p^{12} T^{12} - 376158 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( 1 - 156022 T^{2} + 8400884993 T^{4} + 255692718617586 T^{6} - 56981848883463069212 T^{8} + 255692718617586 p^{6} T^{10} + 8400884993 p^{12} T^{12} - 156022 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 + 450 T + 276753 T^{2} + 75560470 T^{3} + 29002070616 T^{4} + 75560470 p^{3} T^{5} + 276753 p^{6} T^{6} + 450 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 + 47832 T^{2} + 34956403676 T^{4} + 1086958705185768 T^{6} + \)\(51\!\cdots\!98\)\( T^{8} + 1086958705185768 p^{6} T^{10} + 34956403676 p^{12} T^{12} + 47832 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( ( 1 - 262 T + 483789 T^{2} - 119532570 T^{3} + 100466115904 T^{4} - 119532570 p^{3} T^{5} + 483789 p^{6} T^{6} - 262 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( 1 - 1020496 T^{2} + 535237608956 T^{4} - 183778307185645104 T^{6} + \)\(44\!\cdots\!22\)\( T^{8} - 183778307185645104 p^{6} T^{10} + 535237608956 p^{12} T^{12} - 1020496 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 + 928 T + 1041074 T^{2} + 583837824 T^{3} + 364336742755 T^{4} + 583837824 p^{3} T^{5} + 1041074 p^{6} T^{6} + 928 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 1151646 T^{2} + 527220986457 T^{4} - 121357066280803958 T^{6} + \)\(24\!\cdots\!88\)\( T^{8} - 121357066280803958 p^{6} T^{10} + 527220986457 p^{12} T^{12} - 1151646 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( 1 - 1269992 T^{2} + 982709794076 T^{4} - 517443712328228568 T^{6} + \)\(21\!\cdots\!42\)\( T^{8} - 517443712328228568 p^{6} T^{10} + 982709794076 p^{12} T^{12} - 1269992 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( 1 - 629780 T^{2} + 483790183226 T^{4} - 266402828553390384 T^{6} + \)\(10\!\cdots\!75\)\( T^{8} - 266402828553390384 p^{6} T^{10} + 483790183226 p^{12} T^{12} - 629780 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 + 746 T + 2156493 T^{2} + 1122200666 T^{3} + 1640976331028 T^{4} + 1122200666 p^{3} T^{5} + 2156493 p^{6} T^{6} + 746 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 3459568 T^{2} + 5753809741436 T^{4} - 5905413398293656720 T^{6} + \)\(40\!\cdots\!34\)\( T^{8} - 5905413398293656720 p^{6} T^{10} + 5753809741436 p^{12} T^{12} - 3459568 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( 1 - 1387392 T^{2} + 1759988118716 T^{4} - 1699002317027998848 T^{6} + \)\(11\!\cdots\!82\)\( T^{8} - 1699002317027998848 p^{6} T^{10} + 1759988118716 p^{12} T^{12} - 1387392 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( 1 - 37422 p T^{2} + 3401136167793 T^{4} + 2633267699383729330 T^{6} - \)\(63\!\cdots\!04\)\( T^{8} + 2633267699383729330 p^{6} T^{10} + 3401136167793 p^{12} T^{12} - 37422 p^{19} T^{14} + p^{24} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.51301062447735934523132406122, −4.30673810389634406865984052269, −4.20653752096597986898700392987, −4.06289993272182890002597716842, −3.84704005893333192527995394339, −3.83275745072337603431368873003, −3.40269365941492093597327652112, −3.31389404118403436942887140501, −2.99256078026543434940603333478, −2.86661311240178750697275406180, −2.84067014431237230661366244718, −2.65764783847628773666611635176, −2.64764352763160945615163174273, −1.95599148723013546685785741575, −1.95210767810578003530375145422, −1.90802404368299994777597653302, −1.71242492095192423337956334367, −1.61344911550941224133637195850, −1.30796239569681996513018041787, −0.981693491481636556629495009090, −0.971807518913808236602584366253, −0.59132140320739781113644753036, −0.48404589162448832065162214192, −0.46403246498051315348064002905, −0.07239028050315168898227894509,
0.07239028050315168898227894509, 0.46403246498051315348064002905, 0.48404589162448832065162214192, 0.59132140320739781113644753036, 0.971807518913808236602584366253, 0.981693491481636556629495009090, 1.30796239569681996513018041787, 1.61344911550941224133637195850, 1.71242492095192423337956334367, 1.90802404368299994777597653302, 1.95210767810578003530375145422, 1.95599148723013546685785741575, 2.64764352763160945615163174273, 2.65764783847628773666611635176, 2.84067014431237230661366244718, 2.86661311240178750697275406180, 2.99256078026543434940603333478, 3.31389404118403436942887140501, 3.40269365941492093597327652112, 3.83275745072337603431368873003, 3.84704005893333192527995394339, 4.06289993272182890002597716842, 4.20653752096597986898700392987, 4.30673810389634406865984052269, 4.51301062447735934523132406122
Plot not available for L-functions of degree greater than 10.
