| L(s) = 1 | − 8·3-s + 23·4-s − 76·7-s + 91·9-s − 184·12-s + 424·13-s + 265·16-s − 244·19-s + 608·21-s − 440·27-s − 1.74e3·28-s + 3.77e3·31-s + 2.09e3·36-s − 1.89e3·37-s − 3.39e3·39-s + 7.38e3·43-s − 2.12e3·48-s − 4.97e3·49-s + 9.75e3·52-s + 1.95e3·57-s + 6.45e3·61-s − 6.91e3·63-s + 4.70e3·64-s − 1.38e4·67-s − 596·73-s − 5.61e3·76-s − 1.61e4·79-s + ⋯ |
| L(s) = 1 | − 8/9·3-s + 1.43·4-s − 1.55·7-s + 1.12·9-s − 1.27·12-s + 2.50·13-s + 1.03·16-s − 0.675·19-s + 1.37·21-s − 0.603·27-s − 2.22·28-s + 3.92·31-s + 1.61·36-s − 1.38·37-s − 2.23·39-s + 3.99·43-s − 0.920·48-s − 2.07·49-s + 3.60·52-s + 0.600·57-s + 1.73·61-s − 1.74·63-s + 1.14·64-s − 3.07·67-s − 0.111·73-s − 0.971·76-s − 2.58·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.444875312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444875312\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 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 | \( 1 \) |
| good | 2 | \( 1 - 23 T^{2} + 33 p^{3} T^{4} - 1171 p^{2} T^{6} + 33 p^{11} T^{8} - 23 p^{16} T^{10} + p^{24} T^{12} \) |
| 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 | \( ( 1 + 4878 T + 170499663 T^{2} + 362588117636 T^{3} + 170499663 p^{4} T^{4} + 4878 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52914267573519373136016489459, −7.23155864782202735395205466643, −6.84890555984579139929345324492, −6.48370046608426960188769388959, −6.46287431425182438384361439033, −6.36815120953794399196578098274, −6.24591072615474314483471920963, −6.14076687254292460500885206785, −5.70211771835591961805908783157, −5.37970391535560149569048470857, −5.35080871480607929597837353265, −4.72971661392548760921860754846, −4.41675680789601017686674406479, −4.34580003775439538935069239957, −4.00174130661177941993410261402, −3.60538042393554042212640108568, −3.50724959603216408912112844289, −2.84117588555630354603698866456, −2.77180360596670478328353157513, −2.66921253103009499176042821461, −1.95642356976332831520026574265, −1.47839421702366612632555063554, −1.11499446485083801322373491006, −0.996375606195415870720741520045, −0.25574238781681981512856681631,
0.25574238781681981512856681631, 0.996375606195415870720741520045, 1.11499446485083801322373491006, 1.47839421702366612632555063554, 1.95642356976332831520026574265, 2.66921253103009499176042821461, 2.77180360596670478328353157513, 2.84117588555630354603698866456, 3.50724959603216408912112844289, 3.60538042393554042212640108568, 4.00174130661177941993410261402, 4.34580003775439538935069239957, 4.41675680789601017686674406479, 4.72971661392548760921860754846, 5.35080871480607929597837353265, 5.37970391535560149569048470857, 5.70211771835591961805908783157, 6.14076687254292460500885206785, 6.24591072615474314483471920963, 6.36815120953794399196578098274, 6.46287431425182438384361439033, 6.48370046608426960188769388959, 6.84890555984579139929345324492, 7.23155864782202735395205466643, 7.52914267573519373136016489459
Plot not available for L-functions of degree greater than 10.
