| L(s) = 1 | − 8·3-s − 76·7-s + 91·9-s − 424·13-s + 244·19-s + 608·21-s − 375·25-s − 440·27-s − 3.77e3·31-s + 1.89e3·37-s + 3.39e3·39-s + 7.38e3·43-s − 4.97e3·49-s − 1.95e3·57-s + 6.45e3·61-s − 6.91e3·63-s − 1.38e4·67-s + 596·73-s + 3.00e3·75-s + 1.61e4·79-s + 4.13e3·81-s + 3.22e4·91-s + 3.01e4·93-s + 9.75e3·97-s + 2.96e3·103-s − 1.97e4·109-s − 1.51e4·111-s + ⋯ |
| L(s) = 1 | − 8/9·3-s − 1.55·7-s + 1.12·9-s − 2.50·13-s + 0.675·19-s + 1.37·21-s − 3/5·25-s − 0.603·27-s − 3.92·31-s + 1.38·37-s + 2.23·39-s + 3.99·43-s − 2.07·49-s − 0.600·57-s + 1.73·61-s − 1.74·63-s − 3.07·67-s + 0.111·73-s + 8/15·75-s + 2.58·79-s + 0.629·81-s + 3.89·91-s + 3.48·93-s + 1.03·97-s + 0.279·103-s − 1.66·109-s − 1.23·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.004117315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004117315\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 8 T - p^{3} T^{2} - 56 p^{2} T^{3} - p^{7} T^{4} + 8 p^{8} T^{5} + p^{12} T^{6} \) |
| 5 | \( ( 1 + p^{3} T^{2} )^{3} \) |
| good | 7 | \( ( 1 + 38 T + 4653 T^{2} + 114976 T^{3} + 4653 p^{4} T^{4} + 38 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 11 | \( 1 - 34466 T^{2} + 627812895 T^{4} - 9331838759740 T^{6} + 627812895 p^{8} T^{8} - 34466 p^{16} T^{10} + p^{24} T^{12} \) |
| 13 | \( ( 1 + 212 T + 97943 T^{2} + 12289064 T^{3} + 97943 p^{4} T^{4} + 212 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 17 | \( 1 - 256778 T^{2} + 40830133839 T^{4} - 4057376310317644 T^{6} + 40830133839 p^{8} T^{8} - 256778 p^{16} T^{10} + p^{24} T^{12} \) |
| 19 | \( ( 1 - 122 T + 284939 T^{2} - 25213796 T^{3} + 284939 p^{4} T^{4} - 122 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 23 | \( 1 - 541878 T^{2} - 37441234461 T^{4} + 62508481606281076 T^{6} - 37441234461 p^{8} T^{8} - 541878 p^{16} T^{10} + p^{24} T^{12} \) |
| 29 | \( 1 - 1427666 T^{2} + 445361323935 T^{4} + 117229967968878500 T^{6} + 445361323935 p^{8} T^{8} - 1427666 p^{16} T^{10} + p^{24} T^{12} \) |
| 31 | \( ( 1 + 1886 T + 3804195 T^{2} + 3654810940 T^{3} + 3804195 p^{4} T^{4} + 1886 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 37 | \( ( 1 - 948 T + 3630423 T^{2} - 3404430056 T^{3} + 3630423 p^{4} T^{4} - 948 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 41 | \( 1 - 9622286 T^{2} + 49511941482495 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + 49511941482495 p^{8} T^{8} - 9622286 p^{16} T^{10} + p^{24} T^{12} \) |
| 43 | \( ( 1 - 3692 T + 9012053 T^{2} - 15407946584 T^{3} + 9012053 p^{4} T^{4} - 3692 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 47 | \( 1 - 19358678 T^{2} + 191449450821219 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + 191449450821219 p^{8} T^{8} - 19358678 p^{16} T^{10} + p^{24} T^{12} \) |
| 53 | \( 1 - 31471178 T^{2} + 496434526631919 T^{4} - \)\(48\!\cdots\!24\)\( T^{6} + 496434526631919 p^{8} T^{8} - 31471178 p^{16} T^{10} + p^{24} T^{12} \) |
| 59 | \( 1 - 20221586 T^{2} + 518070122195295 T^{4} - 1669548568393377340 p^{2} T^{6} + 518070122195295 p^{8} T^{8} - 20221586 p^{16} T^{10} + p^{24} T^{12} \) |
| 61 | \( ( 1 - 3226 T + 32346515 T^{2} - 81211143220 T^{3} + 32346515 p^{4} T^{4} - 3226 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 67 | \( ( 1 + 6908 T + 65136693 T^{2} + 250302748936 T^{3} + 65136693 p^{4} T^{4} + 6908 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 71 | \( 1 - 121682966 T^{2} + 6612859983587535 T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + 6612859983587535 p^{8} T^{8} - 121682966 p^{16} T^{10} + p^{24} T^{12} \) |
| 73 | \( ( 1 - 298 T + 50688863 T^{2} - 79767435436 T^{3} + 50688863 p^{4} T^{4} - 298 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 79 | \( ( 1 - 8062 T + 112728699 T^{2} - 533077126316 T^{3} + 112728699 p^{4} T^{4} - 8062 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 83 | \( 1 - 211578198 T^{2} + 21276155528463939 T^{4} - \)\(12\!\cdots\!04\)\( T^{6} + 21276155528463939 p^{8} T^{8} - 211578198 p^{16} T^{10} + p^{24} T^{12} \) |
| 89 | \( 1 - 255353766 T^{2} + 32750513998015695 T^{4} - \)\(25\!\cdots\!40\)\( T^{6} + 32750513998015695 p^{8} T^{8} - 255353766 p^{16} T^{10} + p^{24} T^{12} \) |
| 97 | \( ( 1 - 4878 T + 170499663 T^{2} - 362588117636 T^{3} + 170499663 p^{4} T^{4} - 4878 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.03413209820759796429306730379, −5.92039957444893918404244361141, −5.43985699771337750455348786893, −5.30733816288787989409039620615, −5.22359396898258179760922990034, −5.09292333954169983721902926703, −4.68518330374767498737553407910, −4.63505059987487544938547710498, −4.37281628635080703867574937395, −4.03583753127688042012503866248, −3.97980079576240724553351931413, −3.71918334337182706030679171143, −3.48886977480571015415626426899, −3.14761647999325755544980840527, −3.04270325478890121287274620262, −2.77219424928228006875500486936, −2.41628775205237883645708287980, −2.20908018567285924435845027537, −2.01224242853008621339489379620, −1.79545989661179870178733797400, −1.26064673812697185726582087600, −1.10030285572475123777428701382, −0.61267174170352340039287400928, −0.30037588920490447875992941355, −0.25404937739329491432779038090,
0.25404937739329491432779038090, 0.30037588920490447875992941355, 0.61267174170352340039287400928, 1.10030285572475123777428701382, 1.26064673812697185726582087600, 1.79545989661179870178733797400, 2.01224242853008621339489379620, 2.20908018567285924435845027537, 2.41628775205237883645708287980, 2.77219424928228006875500486936, 3.04270325478890121287274620262, 3.14761647999325755544980840527, 3.48886977480571015415626426899, 3.71918334337182706030679171143, 3.97980079576240724553351931413, 4.03583753127688042012503866248, 4.37281628635080703867574937395, 4.63505059987487544938547710498, 4.68518330374767498737553407910, 5.09292333954169983721902926703, 5.22359396898258179760922990034, 5.30733816288787989409039620615, 5.43985699771337750455348786893, 5.92039957444893918404244361141, 6.03413209820759796429306730379
Plot not available for L-functions of degree greater than 10.
