| L(s) = 1 | − 21·4-s + 8.50e4·9-s − 6.88e3·11-s − 1.56e5·16-s − 4.45e5·19-s − 8.16e6·29-s + 5.73e6·31-s − 1.78e6·36-s − 2.88e7·41-s + 1.44e5·44-s − 1.72e7·49-s + 8.51e7·59-s + 3.83e8·61-s + 8.31e7·64-s + 5.93e8·71-s + 9.35e6·76-s + 1.92e9·79-s + 3.85e9·81-s − 1.01e9·89-s − 5.85e8·99-s + 1.05e9·101-s − 6.11e7·109-s + 1.71e8·116-s − 8.75e8·121-s − 1.20e8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.0410·4-s + 4.32·9-s − 0.141·11-s − 0.598·16-s − 0.784·19-s − 2.14·29-s + 1.11·31-s − 0.177·36-s − 1.59·41-s + 0.00581·44-s − 3/7·49-s + 0.915·59-s + 3.54·61-s + 0.619·64-s + 2.76·71-s + 0.0321·76-s + 5.54·79-s + 9.94·81-s − 1.71·89-s − 0.612·99-s + 1.00·101-s − 0.0415·109-s + 0.0879·116-s − 0.371·121-s − 0.0457·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.655976682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655976682\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + p^{8} T^{2} )^{3} \) |
| good | 2 | \( 1 + 21 T^{2} + 39333 p^{2} T^{4} - 1195423 p^{6} T^{6} + 39333 p^{20} T^{8} + 21 p^{36} T^{10} + p^{54} T^{12} \) |
| 3 | \( 1 - 350 p^{5} T^{2} + 13904429 p^{5} T^{4} - 112976497100 p^{6} T^{6} + 13904429 p^{23} T^{8} - 350 p^{41} T^{10} + p^{54} T^{12} \) |
| 11 | \( ( 1 + 3444 T + 455343105 T^{2} + 125101303155960 T^{3} + 455343105 p^{9} T^{4} + 3444 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 13 | \( 1 - 37373102586 T^{2} + \)\(74\!\cdots\!07\)\( T^{4} - \)\(97\!\cdots\!68\)\( T^{6} + \)\(74\!\cdots\!07\)\( p^{18} T^{8} - 37373102586 p^{36} T^{10} + p^{54} T^{12} \) |
| 17 | \( 1 - 264849591930 T^{2} + \)\(40\!\cdots\!07\)\( T^{4} - \)\(50\!\cdots\!40\)\( T^{6} + \)\(40\!\cdots\!07\)\( p^{18} T^{8} - 264849591930 p^{36} T^{10} + p^{54} T^{12} \) |
| 19 | \( ( 1 + 222852 T + 614081373717 T^{2} + 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} + 222852 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 23 | \( 1 - 7279697321610 T^{2} + \)\(26\!\cdots\!87\)\( T^{4} - \)\(60\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!87\)\( p^{18} T^{8} - 7279697321610 p^{36} T^{10} + p^{54} T^{12} \) |
| 29 | \( ( 1 + 4081818 T + 38739015783987 T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} + 4081818 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 31 | \( ( 1 - 2869440 T + 21142500166221 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} - 2869440 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 37 | \( 1 - 523404194786130 T^{2} + \)\(13\!\cdots\!87\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!87\)\( p^{18} T^{8} - 523404194786130 p^{36} T^{10} + p^{54} T^{12} \) |
| 41 | \( ( 1 + 14420658 T + 764979654799959 T^{2} + \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} + 14420658 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 43 | \( 1 - 1576804644236466 T^{2} + \)\(14\!\cdots\!07\)\( T^{4} - \)\(46\!\cdots\!52\)\( p^{2} T^{6} + \)\(14\!\cdots\!07\)\( p^{18} T^{8} - 1576804644236466 p^{36} T^{10} + p^{54} T^{12} \) |
| 47 | \( 1 - 5786138591197818 T^{2} + \)\(14\!\cdots\!07\)\( T^{4} - \)\(21\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!07\)\( p^{18} T^{8} - 5786138591197818 p^{36} T^{10} + p^{54} T^{12} \) |
| 53 | \( 1 - 8033491063762962 T^{2} + \)\(53\!\cdots\!67\)\( T^{4} - \)\(19\!\cdots\!76\)\( T^{6} + \)\(53\!\cdots\!67\)\( p^{18} T^{8} - 8033491063762962 p^{36} T^{10} + p^{54} T^{12} \) |
| 59 | \( ( 1 - 42590100 T + 19076976504365997 T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} - 42590100 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 61 | \( ( 1 - 191746842 T + 38596678668907359 T^{2} - \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 67 | \( 1 - 95230354629887778 T^{2} + \)\(47\!\cdots\!47\)\( T^{4} - \)\(15\!\cdots\!64\)\( T^{6} + \)\(47\!\cdots\!47\)\( p^{18} T^{8} - 95230354629887778 p^{36} T^{10} + p^{54} T^{12} \) |
| 71 | \( ( 1 - 296514504 T + 147895725194380437 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} - 296514504 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 73 | \( 1 - 48113877823008522 T^{2} + \)\(23\!\cdots\!67\)\( T^{4} + \)\(31\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!67\)\( p^{18} T^{8} - 48113877823008522 p^{36} T^{10} + p^{54} T^{12} \) |
| 79 | \( ( 1 - 960412656 T + 566786434394061357 T^{2} - \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} - 960412656 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 83 | \( 1 - 536655338289530202 T^{2} + \)\(14\!\cdots\!27\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!27\)\( p^{18} T^{8} - 536655338289530202 p^{36} T^{10} + p^{54} T^{12} \) |
| 89 | \( ( 1 + 506816478 T + 956194525794688887 T^{2} + \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} + 506816478 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 97 | \( 1 - 1220138505429787098 T^{2} + \)\(12\!\cdots\!67\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!67\)\( p^{18} T^{8} - 1220138505429787098 p^{36} T^{10} + p^{54} T^{12} \) |
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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
−5.29554200955262884200350155590, −4.95257612832066730816012836197, −4.94255316455507408931999102656, −4.67009652998445790268482453126, −4.53264872413193179460655917773, −4.21189683884799826338780450399, −4.02113796465432456286662043056, −3.91515692852679624164496868635, −3.91307490693861259471842603494, −3.45901081972234536979021788085, −3.36463948561674419248831347104, −3.33483757365466571059116675975, −2.75518979994858561130237228565, −2.46103927508604375301394337756, −2.10562849788965077745367250442, −2.09522946859034318198371959893, −1.89170147010155265425600322197, −1.86973545256338023267122411434, −1.69670538405870065630962363110, −1.07632412304607254020876303177, −1.06377572700568057811422880603, −0.866351364759143065827533756803, −0.65959112185435319163336968953, −0.58189767930696070154510353469, −0.05888862680510438162117179853,
0.05888862680510438162117179853, 0.58189767930696070154510353469, 0.65959112185435319163336968953, 0.866351364759143065827533756803, 1.06377572700568057811422880603, 1.07632412304607254020876303177, 1.69670538405870065630962363110, 1.86973545256338023267122411434, 1.89170147010155265425600322197, 2.09522946859034318198371959893, 2.10562849788965077745367250442, 2.46103927508604375301394337756, 2.75518979994858561130237228565, 3.33483757365466571059116675975, 3.36463948561674419248831347104, 3.45901081972234536979021788085, 3.91307490693861259471842603494, 3.91515692852679624164496868635, 4.02113796465432456286662043056, 4.21189683884799826338780450399, 4.53264872413193179460655917773, 4.67009652998445790268482453126, 4.94255316455507408931999102656, 4.95257612832066730816012836197, 5.29554200955262884200350155590
Plot not available for L-functions of degree greater than 10.
