| L(s) = 1 | − 52·3-s − 96·4-s + 368·5-s − 1.00e3·9-s − 1.16e3·11-s + 4.99e3·12-s − 1.91e4·15-s + 6.14e3·16-s − 3.53e4·20-s + 2.15e3·23-s + 5.07e4·25-s + 9.27e4·27-s − 7.84e4·31-s + 6.06e4·33-s + 9.67e4·36-s − 2.05e5·37-s + 1.11e5·44-s − 3.70e5·45-s + 4.93e5·47-s − 3.19e5·48-s + 2.17e5·49-s − 5.31e5·53-s − 4.29e5·55-s − 8.33e5·59-s + 1.83e6·60-s − 3.27e5·64-s + 5.37e5·67-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 3/2·4-s + 2.94·5-s − 1.38·9-s − 0.876·11-s + 26/9·12-s − 5.66·15-s + 3/2·16-s − 4.41·20-s + 0.177·23-s + 3.24·25-s + 4.71·27-s − 2.63·31-s + 1.68·33-s + 2.07·36-s − 4.06·37-s + 1.31·44-s − 4.07·45-s + 4.75·47-s − 2.88·48-s + 1.84·49-s − 3.57·53-s − 2.57·55-s − 4.05·59-s + 8.50·60-s − 5/4·64-s + 1.78·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.06204266188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06204266188\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{5} T^{2} )^{3} \) |
| 11 | \( 1 + 106 p T - 6865 p^{2} T^{2} - 258556 p^{4} T^{3} - 6865 p^{8} T^{4} + 106 p^{13} T^{5} + p^{18} T^{6} \) |
| good | 3 | \( ( 1 + 26 T + 506 p T^{2} + 12290 p T^{3} + 506 p^{7} T^{4} + 26 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 5 | \( ( 1 - 184 T + 1016 p^{2} T^{2} - 68842 p^{2} T^{3} + 1016 p^{8} T^{4} - 184 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 7 | \( 1 - 217566 T^{2} + 32540011983 T^{4} - 4783143212564132 T^{6} + 32540011983 p^{12} T^{8} - 217566 p^{24} T^{10} + p^{36} T^{12} \) |
| 13 | \( 1 - 12318414 T^{2} + 100269018276927 T^{4} - \)\(52\!\cdots\!44\)\( T^{6} + 100269018276927 p^{12} T^{8} - 12318414 p^{24} T^{10} + p^{36} T^{12} \) |
| 17 | \( 1 - 61535358 T^{2} + 1899521995563567 T^{4} - \)\(45\!\cdots\!16\)\( T^{6} + 1899521995563567 p^{12} T^{8} - 61535358 p^{24} T^{10} + p^{36} T^{12} \) |
| 19 | \( 1 - 26275086 T^{2} + 6822221013915327 T^{4} - \)\(11\!\cdots\!76\)\( T^{6} + 6822221013915327 p^{12} T^{8} - 26275086 p^{24} T^{10} + p^{36} T^{12} \) |
| 23 | \( ( 1 - 1078 T + 415875950 T^{2} - 312925939186 T^{3} + 415875950 p^{6} T^{4} - 1078 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 29 | \( 1 - 2760181398 T^{2} + 3539740269101774751 T^{4} - \)\(26\!\cdots\!36\)\( T^{6} + 3539740269101774751 p^{12} T^{8} - 2760181398 p^{24} T^{10} + p^{36} T^{12} \) |
| 31 | \( ( 1 + 39234 T + 2560490982 T^{2} + 68508627283486 T^{3} + 2560490982 p^{6} T^{4} + 39234 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 37 | \( ( 1 + 102960 T + 10729446096 T^{2} + 554698638636454 T^{3} + 10729446096 p^{6} T^{4} + 102960 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 41 | \( 1 - 13389830046 T^{2} + 86823147879328529295 T^{4} - \)\(41\!\cdots\!72\)\( T^{6} + 86823147879328529295 p^{12} T^{8} - 13389830046 p^{24} T^{10} + p^{36} T^{12} \) |
| 43 | \( 1 - 19691349462 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(85\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!31\)\( p^{12} T^{8} - 19691349462 p^{24} T^{10} + p^{36} T^{12} \) |
| 47 | \( ( 1 - 246730 T + 50981717951 T^{2} - 5751342041687596 T^{3} + 50981717951 p^{6} T^{4} - 246730 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 53 | \( ( 1 + 265850 T + 61006376519 T^{2} + 8665677288711308 T^{3} + 61006376519 p^{6} T^{4} + 265850 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 59 | \( ( 1 + 416690 T + 126613451990 T^{2} + 26137140602762390 T^{3} + 126613451990 p^{6} T^{4} + 416690 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 61 | \( 1 - 162819828342 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!15\)\( p^{12} T^{8} - 162819828342 p^{24} T^{10} + p^{36} T^{12} \) |
| 67 | \( ( 1 - 268710 T + 276879935070 T^{2} - 46741488269575706 T^{3} + 276879935070 p^{6} T^{4} - 268710 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 71 | \( ( 1 + 230186 T + 260527573982 T^{2} + 68731919036734094 T^{3} + 260527573982 p^{6} T^{4} + 230186 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 73 | \( 1 - 203158701342 T^{2} + \)\(52\!\cdots\!79\)\( T^{4} - \)\(57\!\cdots\!36\)\( T^{6} + \)\(52\!\cdots\!79\)\( p^{12} T^{8} - 203158701342 p^{24} T^{10} + p^{36} T^{12} \) |
| 79 | \( 1 - 576022933062 T^{2} + \)\(21\!\cdots\!67\)\( T^{4} - \)\(62\!\cdots\!84\)\( T^{6} + \)\(21\!\cdots\!67\)\( p^{12} T^{8} - 576022933062 p^{24} T^{10} + p^{36} T^{12} \) |
| 83 | \( 1 - 217331496918 T^{2} + \)\(22\!\cdots\!43\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(22\!\cdots\!43\)\( p^{12} T^{8} - 217331496918 p^{24} T^{10} + p^{36} T^{12} \) |
| 89 | \( ( 1 - 1188976 T + 1827007541360 T^{2} - 1203827383443407986 T^{3} + 1827007541360 p^{6} T^{4} - 1188976 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 97 | \( ( 1 - 675816 T + 466099093224 T^{2} + 583457088977364286 T^{3} + 466099093224 p^{6} T^{4} - 675816 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
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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
−9.089054287754609490665832589148, −9.016432521789928848506361834727, −8.991742616975132500305818912624, −8.730303738600177487628376083089, −8.320557258802263216212350192508, −7.67058804474121342554499160999, −7.53239047293948819990197998566, −7.42897562937392150075326987693, −6.71049817479345139679322113036, −6.08100850016220483011935435721, −6.07481631404795758352013547369, −5.98620159989822230140865238319, −5.82415986558776311223658135595, −5.29626245506729951983954756978, −5.27850361108800823546382264666, −4.95813472648908245208059018727, −4.89679861497867780779587252285, −3.78887737774675926518631648056, −3.57430447490615509589377678655, −3.00420139107336542270071872918, −2.40066914510890635042382070490, −1.99710007934803792864318452153, −1.55874727887623103869284834703, −0.56662782128803702968651739375, −0.096675761576384289949989446952,
0.096675761576384289949989446952, 0.56662782128803702968651739375, 1.55874727887623103869284834703, 1.99710007934803792864318452153, 2.40066914510890635042382070490, 3.00420139107336542270071872918, 3.57430447490615509589377678655, 3.78887737774675926518631648056, 4.89679861497867780779587252285, 4.95813472648908245208059018727, 5.27850361108800823546382264666, 5.29626245506729951983954756978, 5.82415986558776311223658135595, 5.98620159989822230140865238319, 6.07481631404795758352013547369, 6.08100850016220483011935435721, 6.71049817479345139679322113036, 7.42897562937392150075326987693, 7.53239047293948819990197998566, 7.67058804474121342554499160999, 8.320557258802263216212350192508, 8.730303738600177487628376083089, 8.991742616975132500305818912624, 9.016432521789928848506361834727, 9.089054287754609490665832589148
Plot not available for L-functions of degree greater than 10.
