| L(s) = 1 | + 1.69e3·7-s + 4.18e4·13-s + 3.63e4·19-s + 3.48e5·25-s − 2.05e6·31-s − 9.39e6·37-s + 2.73e6·43-s − 1.40e6·49-s − 4.01e7·61-s − 1.11e8·67-s + 1.28e7·73-s − 2.18e7·79-s + 7.10e7·91-s + 5.37e7·97-s + 6.15e8·103-s + 4.19e8·109-s + 6.29e8·121-s + 127-s + 131-s + 6.17e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.707·7-s + 1.46·13-s + 0.279·19-s + 0.892·25-s − 2.22·31-s − 5.01·37-s + 0.800·43-s − 0.243·49-s − 2.90·61-s − 5.52·67-s + 0.451·73-s − 0.560·79-s + 1.03·91-s + 0.606·97-s + 5.46·103-s + 2.97·109-s + 2.93·121-s + 0.197·133-s − 3.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5599985039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5599985039\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 348531 T^{2} + 4287300918 p^{2} T^{4} - 83617663627751 p^{4} T^{6} + 4287300918 p^{18} T^{8} - 348531 p^{32} T^{10} + p^{48} T^{12} \) |
| 7 | \( ( 1 - 849 T + 254634 p T^{2} - 2229277 p^{2} T^{3} + 254634 p^{9} T^{4} - 849 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 11 | \( 1 - 629247147 T^{2} + 146743358212989654 T^{4} - \)\(24\!\cdots\!71\)\( T^{6} + 146743358212989654 p^{16} T^{8} - 629247147 p^{32} T^{10} + p^{48} T^{12} \) |
| 13 | \( ( 1 - 20922 T + 2222863503 T^{2} - 30819103063724 T^{3} + 2222863503 p^{8} T^{4} - 20922 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 17 | \( 1 - 18964363578 T^{2} + \)\(22\!\cdots\!31\)\( T^{4} - \)\(19\!\cdots\!72\)\( T^{6} + \)\(22\!\cdots\!31\)\( p^{16} T^{8} - 18964363578 p^{32} T^{10} + p^{48} T^{12} \) |
| 19 | \( ( 1 - 18192 T + 47345904771 T^{2} - 35368548610912 p T^{3} + 47345904771 p^{8} T^{4} - 18192 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 23 | \( 1 - 229542851094 T^{2} + 58965330274736482527 p^{2} T^{4} - \)\(10\!\cdots\!60\)\( p^{4} T^{6} + 58965330274736482527 p^{18} T^{8} - 229542851094 p^{32} T^{10} + p^{48} T^{12} \) |
| 29 | \( 1 - 893975304282 T^{2} + \)\(61\!\cdots\!19\)\( T^{4} + \)\(14\!\cdots\!64\)\( T^{6} + \)\(61\!\cdots\!19\)\( p^{16} T^{8} - 893975304282 p^{32} T^{10} + p^{48} T^{12} \) |
| 31 | \( ( 1 + 1029237 T + 1267759379886 T^{2} + 29664245756200343 p T^{3} + 1267759379886 p^{8} T^{4} + 1029237 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 37 | \( ( 1 + 4697940 T + 17009082078963 T^{2} + 35283698248403317480 T^{3} + 17009082078963 p^{8} T^{4} + 4697940 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 41 | \( 1 - 18537322805142 T^{2} + \)\(22\!\cdots\!59\)\( T^{4} - \)\(21\!\cdots\!56\)\( T^{6} + \)\(22\!\cdots\!59\)\( p^{16} T^{8} - 18537322805142 p^{32} T^{10} + p^{48} T^{12} \) |
| 43 | \( ( 1 - 1368642 T + 14910735053103 T^{2} - 33425069167473118684 T^{3} + 14910735053103 p^{8} T^{4} - 1368642 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 47 | \( 1 - 53523898933722 T^{2} + \)\(24\!\cdots\!63\)\( T^{4} - \)\(63\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!63\)\( p^{16} T^{8} - 53523898933722 p^{32} T^{10} + p^{48} T^{12} \) |
| 53 | \( 1 - 275932922945955 T^{2} + \)\(36\!\cdots\!10\)\( T^{4} - \)\(10\!\cdots\!87\)\( p^{2} T^{6} + \)\(36\!\cdots\!10\)\( p^{16} T^{8} - 275932922945955 p^{32} T^{10} + p^{48} T^{12} \) |
| 59 | \( 1 - 618106182519642 T^{2} + \)\(19\!\cdots\!59\)\( T^{4} - \)\(35\!\cdots\!56\)\( T^{6} + \)\(19\!\cdots\!59\)\( p^{16} T^{8} - 618106182519642 p^{32} T^{10} + p^{48} T^{12} \) |
| 61 | \( ( 1 + 20090388 T + 428254629112851 T^{2} + \)\(46\!\cdots\!52\)\( T^{3} + 428254629112851 p^{8} T^{4} + 20090388 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 67 | \( ( 1 + 55677642 T + 1719460921575183 T^{2} + \)\(37\!\cdots\!44\)\( T^{3} + 1719460921575183 p^{8} T^{4} + 55677642 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 71 | \( 1 - 1912238145877722 T^{2} + \)\(19\!\cdots\!79\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(19\!\cdots\!79\)\( p^{16} T^{8} - 1912238145877722 p^{32} T^{10} + p^{48} T^{12} \) |
| 73 | \( ( 1 - 6410859 T + 1962800745716598 T^{2} - \)\(60\!\cdots\!83\)\( T^{3} + 1962800745716598 p^{8} T^{4} - 6410859 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 79 | \( ( 1 + 10910202 T + 3779652360873039 T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + 3779652360873039 p^{8} T^{4} + 10910202 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
| 83 | \( 1 - 7120116563607555 T^{2} + \)\(24\!\cdots\!10\)\( T^{4} - \)\(58\!\cdots\!03\)\( T^{6} + \)\(24\!\cdots\!10\)\( p^{16} T^{8} - 7120116563607555 p^{32} T^{10} + p^{48} T^{12} \) |
| 89 | \( 1 - 20713170292850202 T^{2} + \)\(18\!\cdots\!99\)\( T^{4} - \)\(96\!\cdots\!76\)\( T^{6} + \)\(18\!\cdots\!99\)\( p^{16} T^{8} - 20713170292850202 p^{32} T^{10} + p^{48} T^{12} \) |
| 97 | \( ( 1 - 26867553 T + 240239385500334 p T^{2} - \)\(41\!\cdots\!41\)\( T^{3} + 240239385500334 p^{9} T^{4} - 26867553 p^{16} T^{5} + p^{24} T^{6} )^{2} \) |
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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
−4.74769715513142463879594780780, −4.43984228106724144567601983547, −4.25541854912170601423747303243, −4.22274015711246309151321766071, −4.09953300164527471970018626865, −3.57763793348940339813923537337, −3.41811148265853697463198894302, −3.34896703910948810059800715797, −3.30956133203406127081748845594, −3.17854221403345918907948872962, −3.10495710956699362870596398931, −2.73647649748508919686605202116, −2.27635637675520159473656984876, −2.14292286612781842633393503943, −2.14012385078072196937856963418, −1.72696081184901965926157209338, −1.68361779633952009715377841994, −1.57029894761834394802507331687, −1.45825432224840532622075349340, −1.08155261834890182808564752725, −1.02579702851793430069870774120, −0.73173403958300050984284672904, −0.50326072707747431359087672044, −0.15308412943508015779079051678, −0.092625370636652013999604552437,
0.092625370636652013999604552437, 0.15308412943508015779079051678, 0.50326072707747431359087672044, 0.73173403958300050984284672904, 1.02579702851793430069870774120, 1.08155261834890182808564752725, 1.45825432224840532622075349340, 1.57029894761834394802507331687, 1.68361779633952009715377841994, 1.72696081184901965926157209338, 2.14012385078072196937856963418, 2.14292286612781842633393503943, 2.27635637675520159473656984876, 2.73647649748508919686605202116, 3.10495710956699362870596398931, 3.17854221403345918907948872962, 3.30956133203406127081748845594, 3.34896703910948810059800715797, 3.41811148265853697463198894302, 3.57763793348940339813923537337, 4.09953300164527471970018626865, 4.22274015711246309151321766071, 4.25541854912170601423747303243, 4.43984228106724144567601983547, 4.74769715513142463879594780780
Plot not available for L-functions of degree greater than 10.
