| L(s) = 1 | − 144·7-s + 276·13-s − 768·16-s − 5.85e4·31-s − 2.57e4·37-s − 1.60e4·43-s + 1.03e4·49-s − 1.45e5·61-s − 3.35e4·67-s + 1.58e5·73-s − 3.97e4·91-s + 6.31e5·97-s − 2.03e5·103-s + 1.10e5·112-s + 1.84e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.80e4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.11·7-s + 0.452·13-s − 3/4·16-s − 10.9·31-s − 3.09·37-s − 1.32·43-s + 0.616·49-s − 4.99·61-s − 0.913·67-s + 3.49·73-s − 0.503·91-s + 6.81·97-s − 1.88·103-s + 0.833·112-s + 1.14·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.102·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5220789928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5220789928\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{8} T^{4} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 72 T + 2592 T^{2} - 1653800 T^{3} - 278157165 T^{4} + 13719293232 T^{5} + 3076299704384 T^{6} + 13719293232 p^{5} T^{7} - 278157165 p^{10} T^{8} - 1653800 p^{15} T^{9} + 2592 p^{20} T^{10} + 72 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 11 | \( ( 1 - 92322 T^{2} + 23752309431 T^{4} + 14324197252 p^{4} T^{6} + 23752309431 p^{10} T^{8} - 92322 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 13 | \( ( 1 - 138 T + 9522 T^{2} - 71250978 T^{3} + 207133277751 T^{4} + 9625164888756 T^{5} - 762244892415108 T^{6} + 9625164888756 p^{5} T^{7} + 207133277751 p^{10} T^{8} - 71250978 p^{15} T^{9} + 9522 p^{20} T^{10} - 138 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 17 | \( 1 - 3104114865402 T^{4} + \)\(11\!\cdots\!71\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{12} + \)\(11\!\cdots\!71\)\( p^{20} T^{16} - 3104114865402 p^{40} T^{20} + p^{60} T^{24} \) |
| 19 | \( ( 1 - 11705682 T^{2} + 61048847304711 T^{4} - \)\(18\!\cdots\!48\)\( T^{6} + 61048847304711 p^{10} T^{8} - 11705682 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 23 | \( 1 - 54021755287194 T^{4} + \)\(80\!\cdots\!15\)\( T^{8} - \)\(23\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!15\)\( p^{20} T^{16} - 54021755287194 p^{40} T^{20} + p^{60} T^{24} \) |
| 29 | \( ( 1 + 23780088 T^{2} + 922477841694651 T^{4} + \)\(20\!\cdots\!12\)\( T^{6} + 922477841694651 p^{10} T^{8} + 23780088 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 31 | \( ( 1 + 14628 T + 134742381 T^{2} + 812936587832 T^{3} + 134742381 p^{5} T^{4} + 14628 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 37 | \( ( 1 + 12882 T + 82972962 T^{2} + 847667990010 T^{3} + 1418198677343175 T^{4} - 36911476364572499268 T^{5} - \)\(23\!\cdots\!76\)\( T^{6} - 36911476364572499268 p^{5} T^{7} + 1418198677343175 p^{10} T^{8} + 847667990010 p^{15} T^{9} + 82972962 p^{20} T^{10} + 12882 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 41 | \( ( 1 - 675414192 T^{2} + 192260052740082291 T^{4} - \)\(29\!\cdots\!28\)\( T^{6} + 192260052740082291 p^{10} T^{8} - 675414192 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 43 | \( ( 1 + 8040 T + 32320800 T^{2} + 1027788105720 T^{3} + 39748209890076747 T^{4} + 4806251738124883440 p T^{5} + \)\(48\!\cdots\!00\)\( p^{2} T^{6} + 4806251738124883440 p^{6} T^{7} + 39748209890076747 p^{10} T^{8} + 1027788105720 p^{15} T^{9} + 32320800 p^{20} T^{10} + 8040 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 47 | \( 1 - 44718551206350906 T^{4} - \)\(15\!\cdots\!85\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{12} - \)\(15\!\cdots\!85\)\( p^{20} T^{16} - 44718551206350906 p^{40} T^{20} + p^{60} T^{24} \) |
| 53 | \( 1 + 138029827723659606 T^{4} + \)\(13\!\cdots\!15\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{12} + \)\(13\!\cdots\!15\)\( p^{20} T^{16} + 138029827723659606 p^{40} T^{20} + p^{60} T^{24} \) |
| 59 | \( ( 1 + 2197871970 T^{2} + 2372494875416951703 T^{4} + \)\(52\!\cdots\!40\)\( p^{2} T^{6} + 2372494875416951703 p^{10} T^{8} + 2197871970 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 61 | \( ( 1 + 36300 T + 1780029303 T^{2} + 33645024252600 T^{3} + 1780029303 p^{5} T^{4} + 36300 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 67 | \( ( 1 + 16776 T + 140717088 T^{2} + 71734444130584 T^{3} - 1822301007087929349 T^{4} - \)\(12\!\cdots\!32\)\( T^{5} + \)\(71\!\cdots\!08\)\( T^{6} - \)\(12\!\cdots\!32\)\( p^{5} T^{7} - 1822301007087929349 p^{10} T^{8} + 71734444130584 p^{15} T^{9} + 140717088 p^{20} T^{10} + 16776 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 71 | \( ( 1 - 9374370570 T^{2} + 38765983187289347103 T^{4} - \)\(90\!\cdots\!40\)\( T^{6} + 38765983187289347103 p^{10} T^{8} - 9374370570 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 73 | \( ( 1 - 79494 T + 3159648018 T^{2} + 50403657969290 T^{3} - 1967048070000525585 T^{4} - \)\(27\!\cdots\!24\)\( T^{5} + \)\(29\!\cdots\!36\)\( T^{6} - \)\(27\!\cdots\!24\)\( p^{5} T^{7} - 1967048070000525585 p^{10} T^{8} + 50403657969290 p^{15} T^{9} + 3159648018 p^{20} T^{10} - 79494 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
| 79 | \( ( 1 - 12913071402 T^{2} + 82695789000125977071 T^{4} - \)\(31\!\cdots\!08\)\( T^{6} + 82695789000125977071 p^{10} T^{8} - 12913071402 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 83 | \( 1 + 22486620193935462198 T^{4} + \)\(77\!\cdots\!71\)\( T^{8} + \)\(10\!\cdots\!92\)\( T^{12} + \)\(77\!\cdots\!71\)\( p^{20} T^{16} + 22486620193935462198 p^{40} T^{20} + p^{60} T^{24} \) |
| 89 | \( ( 1 + 5060350560 T^{2} + 16753820650674742803 T^{4} - \)\(12\!\cdots\!80\)\( T^{6} + 16753820650674742803 p^{10} T^{8} + 5060350560 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 97 | \( ( 1 - 315558 T + 49788425682 T^{2} - 5316509668950630 T^{3} + \)\(47\!\cdots\!15\)\( T^{4} - \)\(43\!\cdots\!68\)\( T^{5} + \)\(40\!\cdots\!64\)\( T^{6} - \)\(43\!\cdots\!68\)\( p^{5} T^{7} + \)\(47\!\cdots\!15\)\( p^{10} T^{8} - 5316509668950630 p^{15} T^{9} + 49788425682 p^{20} T^{10} - 315558 p^{25} T^{11} + p^{30} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.95604425718516036784586563074, −2.85346828144985156025266134281, −2.70424971341060759690028632107, −2.26781757255347584371174699904, −2.17750913046469364878128946265, −2.13741889986017680872490765193, −2.12230973178838414706131495527, −1.95538937773686063225321413526, −1.83350524718394051788344636848, −1.80465609594179232276923259308, −1.77765555028805847237027635585, −1.75225578686905416686050102301, −1.68665609795628361731656727475, −1.55855582805331705654976764003, −1.42634111648411637478629161953, −1.15430848338211221797008039500, −0.872716406403190245542588900573, −0.858434406296380157478807169762, −0.64766646985999600484168286756, −0.59897167423450324057715138052, −0.59719697486085480500088880203, −0.30905470996921978372044393143, −0.27078184928953055793804800910, −0.12624230548712318947021804397, −0.084253886650862377925148214188,
0.084253886650862377925148214188, 0.12624230548712318947021804397, 0.27078184928953055793804800910, 0.30905470996921978372044393143, 0.59719697486085480500088880203, 0.59897167423450324057715138052, 0.64766646985999600484168286756, 0.858434406296380157478807169762, 0.872716406403190245542588900573, 1.15430848338211221797008039500, 1.42634111648411637478629161953, 1.55855582805331705654976764003, 1.68665609795628361731656727475, 1.75225578686905416686050102301, 1.77765555028805847237027635585, 1.80465609594179232276923259308, 1.83350524718394051788344636848, 1.95538937773686063225321413526, 2.12230973178838414706131495527, 2.13741889986017680872490765193, 2.17750913046469364878128946265, 2.26781757255347584371174699904, 2.70424971341060759690028632107, 2.85346828144985156025266134281, 2.95604425718516036784586563074
Plot not available for L-functions of degree greater than 10.
