| 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 − 375·25-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 − 3.00e3·75-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 − 3/5·25-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 − 0.533·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11390625 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11390625 ^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.710605290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.710605290\) |
| \(L(3)\) |
|
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 | \( ( 1 + p^{3} T^{2} )^{3} \) |
| 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
−10.94255400681828369255585473219, −10.65449381650564773982663754369, −10.10374022320347153386086607773, −9.907941098415344876245097753995, −9.830130373738813280646287040215, −9.808552419872132758236017975395, −9.428111401023613053448076948622, −8.436643640357143837407988803869, −8.314438726321527638568650041886, −8.269509247895973920460388757882, −8.234456067188329212862238679530, −7.46947561328559555476887595680, −7.42894828303382974101147428451, −7.01388605599170706138399106500, −6.54401160512865343367343061001, −6.51769906653217427241821928569, −6.02178243696339248007025197281, −5.04728743721129814869210372854, −4.92795468133220458975084878699, −4.78395746241869307537520082550, −4.12723968844093698482789691866, −3.32535912491251712982761644297, −2.55489415175645136772515103723, −2.31137887005096632444979580234, −1.54783535168467047008664143431,
1.54783535168467047008664143431, 2.31137887005096632444979580234, 2.55489415175645136772515103723, 3.32535912491251712982761644297, 4.12723968844093698482789691866, 4.78395746241869307537520082550, 4.92795468133220458975084878699, 5.04728743721129814869210372854, 6.02178243696339248007025197281, 6.51769906653217427241821928569, 6.54401160512865343367343061001, 7.01388605599170706138399106500, 7.42894828303382974101147428451, 7.46947561328559555476887595680, 8.234456067188329212862238679530, 8.269509247895973920460388757882, 8.314438726321527638568650041886, 8.436643640357143837407988803869, 9.428111401023613053448076948622, 9.808552419872132758236017975395, 9.830130373738813280646287040215, 9.907941098415344876245097753995, 10.10374022320347153386086607773, 10.65449381650564773982663754369, 10.94255400681828369255585473219
Plot not available for L-functions of degree greater than 10.
