| L(s) = 1 | + 8·3-s + 12·7-s + 32·9-s − 120·13-s − 48·16-s + 96·21-s + 252·25-s − 144·27-s − 504·31-s + 768·37-s − 960·39-s − 1.96e3·43-s − 384·48-s + 72·49-s + 1.84e3·61-s + 384·63-s − 1.75e3·67-s + 180·73-s + 2.01e3·75-s − 2.74e3·81-s − 1.44e3·91-s − 4.03e3·93-s − 7.59e3·97-s − 108·103-s + 6.14e3·111-s − 576·112-s − 3.84e3·117-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.647·7-s + 1.18·9-s − 2.56·13-s − 3/4·16-s + 0.997·21-s + 2.01·25-s − 1.02·27-s − 2.92·31-s + 3.41·37-s − 3.94·39-s − 6.97·43-s − 1.15·48-s + 0.209·49-s + 3.87·61-s + 0.767·63-s − 3.19·67-s + 0.288·73-s + 3.10·75-s − 3.76·81-s − 1.65·91-s − 4.49·93-s − 7.95·97-s − 0.103·103-s + 5.25·111-s − 0.485·112-s − 3.03·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.118702178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118702178\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{4} )^{3} \) |
| 3 | \( 1 - 8 T + 32 T^{2} + 16 p^{2} T^{3} - 65 p^{2} T^{4} - 392 p^{2} T^{5} + 6368 p^{2} T^{6} - 392 p^{5} T^{7} - 65 p^{8} T^{8} + 16 p^{11} T^{9} + 32 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 - 252 T^{2} + 6867 p T^{4} - 143128 p^{2} T^{6} + 6867 p^{7} T^{8} - 252 p^{12} T^{10} + p^{18} T^{12} \) |
| good | 7 | \( ( 1 - 6 T + 18 T^{2} - 5338 T^{3} - 56145 T^{4} + 425892 p T^{5} - 53668 p^{2} T^{6} + 425892 p^{4} T^{7} - 56145 p^{6} T^{8} - 5338 p^{9} T^{9} + 18 p^{12} T^{10} - 6 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 11 | \( ( 1 - 5160 T^{2} + 12910383 T^{4} - 20646367520 T^{6} + 12910383 p^{6} T^{8} - 5160 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 13 | \( ( 1 + 60 T + 1800 T^{2} + 158820 T^{3} - 3490473 T^{4} - 527143920 T^{5} - 12733887600 T^{6} - 527143920 p^{3} T^{7} - 3490473 p^{6} T^{8} + 158820 p^{9} T^{9} + 1800 p^{12} T^{10} + 60 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 17 | \( 1 + 49389246 T^{4} + 2350897710158655 T^{8} + \)\(57\!\cdots\!80\)\( T^{12} + 2350897710158655 p^{12} T^{16} + 49389246 p^{24} T^{20} + p^{36} T^{24} \) |
| 19 | \( ( 1 - 19866 T^{2} + 170639175 T^{4} - 1104897765100 T^{6} + 170639175 p^{6} T^{8} - 19866 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 23 | \( 1 + 312679326 T^{4} + 64468656143076255 T^{8} + \)\(93\!\cdots\!80\)\( T^{12} + 64468656143076255 p^{12} T^{16} + 312679326 p^{24} T^{20} + p^{36} T^{24} \) |
| 29 | \( ( 1 + 64164 T^{2} + 2591834895 T^{4} + 79097799142760 T^{6} + 2591834895 p^{6} T^{8} + 64164 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 31 | \( ( 1 + 126 T + 82365 T^{2} + 7546820 T^{3} + 82365 p^{3} T^{4} + 126 p^{6} T^{5} + p^{9} T^{6} )^{4} \) |
| 37 | \( ( 1 - 384 T + 73728 T^{2} - 22020072 T^{3} + 4952550855 T^{4} - 563616222744 T^{5} + 93728745538848 T^{6} - 563616222744 p^{3} T^{7} + 4952550855 p^{6} T^{8} - 22020072 p^{9} T^{9} + 73728 p^{12} T^{10} - 384 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 41 | \( ( 1 - 236790 T^{2} + 22994731743 T^{4} - 1574683574676980 T^{6} + 22994731743 p^{6} T^{8} - 236790 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 43 | \( ( 1 + 984 T + 484128 T^{2} + 201821808 T^{3} + 79971224487 T^{4} + 26341039135032 T^{5} + 7569294632623584 T^{6} + 26341039135032 p^{3} T^{7} + 79971224487 p^{6} T^{8} + 201821808 p^{9} T^{9} + 484128 p^{12} T^{10} + 984 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 47 | \( 1 - 4319847426 T^{4} - 94093646960564686785 T^{8} + \)\(57\!\cdots\!80\)\( T^{12} - 94093646960564686785 p^{12} T^{16} - 4319847426 p^{24} T^{20} + p^{36} T^{24} \) |
| 53 | \( 1 - 6102986514 T^{4} - \)\(11\!\cdots\!45\)\( T^{8} + \)\(21\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!45\)\( p^{12} T^{16} - 6102986514 p^{24} T^{20} + p^{36} T^{24} \) |
| 59 | \( ( 1 + 637440 T^{2} + 226730664303 T^{4} + 55317930376368080 T^{6} + 226730664303 p^{6} T^{8} + 637440 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 61 | \( ( 1 - 462 T + 361611 T^{2} - 85975188 T^{3} + 361611 p^{3} T^{4} - 462 p^{6} T^{5} + p^{9} T^{6} )^{4} \) |
| 67 | \( ( 1 + 876 T + 383688 T^{2} + 289223708 T^{3} + 285344946615 T^{4} + 147185650836576 T^{5} + 61276374890659088 T^{6} + 147185650836576 p^{3} T^{7} + 285344946615 p^{6} T^{8} + 289223708 p^{9} T^{9} + 383688 p^{12} T^{10} + 876 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 71 | \( ( 1 - 1964586 T^{2} + 1663488367695 T^{4} - 779125513757104940 T^{6} + 1663488367695 p^{6} T^{8} - 1964586 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 73 | \( ( 1 - 90 T + 4050 T^{2} + 352652470 T^{3} - 129826243953 T^{4} - 68650192382220 T^{5} + 68886195900959900 T^{6} - 68650192382220 p^{3} T^{7} - 129826243953 p^{6} T^{8} + 352652470 p^{9} T^{9} + 4050 p^{12} T^{10} - 90 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 79 | \( ( 1 - 1686366 T^{2} + 1396022652495 T^{4} - 794062506512918180 T^{6} + 1396022652495 p^{6} T^{8} - 1686366 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 83 | \( 1 + 459606477198 T^{4} - \)\(10\!\cdots\!69\)\( T^{8} - \)\(95\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!69\)\( p^{12} T^{16} + 459606477198 p^{24} T^{20} + p^{36} T^{24} \) |
| 89 | \( ( 1 + 3686310 T^{2} + 6017120586063 T^{4} + 5515082362513072820 T^{6} + 6017120586063 p^{6} T^{8} + 3686310 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 97 | \( ( 1 + 3798 T + 7212402 T^{2} + 10874723094 T^{3} + 14489430094527 T^{4} + 16106501995923924 T^{5} + 15798701173480805916 T^{6} + 16106501995923924 p^{3} T^{7} + 14489430094527 p^{6} T^{8} + 10874723094 p^{9} T^{9} + 7212402 p^{12} T^{10} + 3798 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
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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
−6.09458833311637835187789825201, −5.86062363222426949297947909647, −5.72847902029434680423840119735, −5.68727237491536995151870098447, −5.59079399578043611699415218112, −5.51938868198844940233480007351, −5.00271259326057183984989015025, −4.88016124865950638104623870503, −4.85800779309650030496498425682, −4.73115582973570092952389375873, −4.62941009331432254602558461799, −4.37231136210278044853943809195, −4.14391508224766006809114864555, −3.82810423767403620997095334860, −3.59072417401754754041394698605, −3.56481770583150076019208709715, −3.34510450281509766366314374369, −2.93278997155875450581078334757, −2.64009263077471098614936880808, −2.63684247479145437189268047591, −2.44534059872349672904831074703, −1.88207527276575017981406563618, −1.64880579464028807044679322422, −1.52379729157583607983799455237, −0.24086295526618307626386041841,
0.24086295526618307626386041841, 1.52379729157583607983799455237, 1.64880579464028807044679322422, 1.88207527276575017981406563618, 2.44534059872349672904831074703, 2.63684247479145437189268047591, 2.64009263077471098614936880808, 2.93278997155875450581078334757, 3.34510450281509766366314374369, 3.56481770583150076019208709715, 3.59072417401754754041394698605, 3.82810423767403620997095334860, 4.14391508224766006809114864555, 4.37231136210278044853943809195, 4.62941009331432254602558461799, 4.73115582973570092952389375873, 4.85800779309650030496498425682, 4.88016124865950638104623870503, 5.00271259326057183984989015025, 5.51938868198844940233480007351, 5.59079399578043611699415218112, 5.68727237491536995151870098447, 5.72847902029434680423840119735, 5.86062363222426949297947909647, 6.09458833311637835187789825201
Plot not available for L-functions of degree greater than 10.
