| L(s) = 1 | − 8·4-s − 84·13-s − 25·16-s + 12·19-s − 296·25-s + 800·31-s + 692·37-s + 1.45e3·43-s + 672·52-s + 216·61-s + 990·64-s + 684·67-s − 4.56e3·73-s − 96·76-s + 556·79-s + 584·97-s + 2.36e3·100-s + 3.22e3·103-s + 3.84e3·109-s − 2.36e3·121-s − 6.40e3·124-s + 127-s + 131-s + 137-s + 139-s − 5.53e3·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 1.79·13-s − 0.390·16-s + 0.144·19-s − 2.36·25-s + 4.63·31-s + 3.07·37-s + 5.16·43-s + 1.79·52-s + 0.453·61-s + 1.93·64-s + 1.24·67-s − 7.31·73-s − 0.144·76-s + 0.791·79-s + 0.611·97-s + 2.36·100-s + 3.08·103-s + 3.38·109-s − 1.77·121-s − 4.63·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.07·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7336665226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7336665226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} + 89 T^{4} - 39 p T^{6} - 73 p^{2} T^{8} - 39 p^{7} T^{10} + 89 p^{12} T^{12} + p^{21} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 + 296 T^{2} + 29672 T^{4} - 525912 T^{6} - 390561682 T^{8} - 525912 p^{6} T^{10} + 29672 p^{12} T^{12} + 296 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 + 2360 T^{2} + 7703996 T^{4} + 11981189832 T^{6} + 21079156257062 T^{8} + 11981189832 p^{6} T^{10} + 7703996 p^{12} T^{12} + 2360 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( ( 1 + 42 T + 4313 T^{2} + 182010 T^{3} + 14557656 T^{4} + 182010 p^{3} T^{5} + 4313 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 17 | \( 1 + 5200 T^{2} + 54407944 T^{4} + 410772413536 T^{6} + 1483567725728974 T^{8} + 410772413536 p^{6} T^{10} + 54407944 p^{12} T^{12} + 5200 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 6 T + 12965 T^{2} + 738042 T^{3} + 71871180 T^{4} + 738042 p^{3} T^{5} + 12965 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 + 76840 T^{2} + 2761902568 T^{4} + 60766751914504 T^{6} + 893967887430461902 T^{8} + 60766751914504 p^{6} T^{10} + 2761902568 p^{12} T^{12} + 76840 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( 1 + 64112 T^{2} + 2244902696 T^{4} + 74829707081088 T^{6} + 2150145485373131438 T^{8} + 74829707081088 p^{6} T^{10} + 2244902696 p^{12} T^{12} + 64112 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 - 400 T + 133022 T^{2} - 29046888 T^{3} + 5807114183 T^{4} - 29046888 p^{3} T^{5} + 133022 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 346 T + 200009 T^{2} - 45146394 T^{3} + 14725259912 T^{4} - 45146394 p^{3} T^{5} + 200009 p^{6} T^{6} - 346 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( 1 + 322928 T^{2} + 52903929992 T^{4} + 5861879045254080 T^{6} + \)\(47\!\cdots\!58\)\( T^{8} + 5861879045254080 p^{6} T^{10} + 52903929992 p^{12} T^{12} + 322928 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 - 728 T + 300862 T^{2} - 84588392 T^{3} + 23173084231 T^{4} - 84588392 p^{3} T^{5} + 300862 p^{6} T^{6} - 728 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 + 393280 T^{2} + 76760617960 T^{4} + 10087452510506512 T^{6} + \)\(10\!\cdots\!10\)\( T^{8} + 10087452510506512 p^{6} T^{10} + 76760617960 p^{12} T^{12} + 393280 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 + 471976 T^{2} + 142137435016 T^{4} + 28544169475512808 T^{6} + \)\(47\!\cdots\!86\)\( T^{8} + 28544169475512808 p^{6} T^{10} + 142137435016 p^{12} T^{12} + 471976 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 + 658456 T^{2} + 289747003708 T^{4} + 84023879294730472 T^{6} + \)\(19\!\cdots\!18\)\( T^{8} + 84023879294730472 p^{6} T^{10} + 289747003708 p^{12} T^{12} + 658456 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 - 108 T + 803738 T^{2} - 50204160 T^{3} + 260511323619 T^{4} - 50204160 p^{3} T^{5} + 803738 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 342 T + 682769 T^{2} - 68101038 T^{3} + 212950851600 T^{4} - 68101038 p^{3} T^{5} + 682769 p^{6} T^{6} - 342 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( 1 + 1215712 T^{2} + 723512460412 T^{4} + 361495508434948384 T^{6} + \)\(15\!\cdots\!06\)\( T^{8} + 361495508434948384 p^{6} T^{10} + 723512460412 p^{12} T^{12} + 1215712 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( ( 1 + 2282 T + 2801765 T^{2} + 2298829050 T^{3} + 1547163369020 T^{4} + 2298829050 p^{3} T^{5} + 2801765 p^{6} T^{6} + 2282 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 278 T + 1695313 T^{2} - 313665446 T^{3} + 1182581506636 T^{4} - 313665446 p^{3} T^{5} + 1695313 p^{6} T^{6} - 278 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 + 3328912 T^{2} + 5266809982972 T^{4} + 5198503152929577712 T^{6} + \)\(35\!\cdots\!74\)\( T^{8} + 5198503152929577712 p^{6} T^{10} + 5266809982972 p^{12} T^{12} + 3328912 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( 1 + 5344040 T^{2} + 12678561760136 T^{4} + 17454562317746156328 T^{6} + \)\(15\!\cdots\!62\)\( T^{8} + 17454562317746156328 p^{6} T^{10} + 12678561760136 p^{12} T^{12} + 5344040 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( ( 1 - 292 T + 2972006 T^{2} - 740181864 T^{3} + 3859311404039 T^{4} - 740181864 p^{3} T^{5} + 2972006 p^{6} T^{6} - 292 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.69903853925837043506586809372, −3.43425060383822840895468536265, −3.30956944547911110256672801830, −3.28124396318783369939309500597, −3.26254766096715141907415696032, −2.96856290998462054614309181260, −2.94625236814399587919780433212, −2.65684915931880029178794074640, −2.41493247290483231253852739880, −2.38860418366126324414697801934, −2.38256583256639514564599558328, −2.36133750393549667424038340583, −2.31512589412499989767793667148, −2.07839319178881540484981670390, −1.65110876314222425938332222477, −1.60391730887979119155092026682, −1.38057079104127782201852403787, −1.27134869292084255218831776704, −1.03493329420050309623928340080, −0.828508359095859483495065023210, −0.74133009149643687728036443095, −0.58950931355853476579167682724, −0.57250992815341495652419694793, −0.33833020353531410404616194610, −0.04865050536869086106741872234,
0.04865050536869086106741872234, 0.33833020353531410404616194610, 0.57250992815341495652419694793, 0.58950931355853476579167682724, 0.74133009149643687728036443095, 0.828508359095859483495065023210, 1.03493329420050309623928340080, 1.27134869292084255218831776704, 1.38057079104127782201852403787, 1.60391730887979119155092026682, 1.65110876314222425938332222477, 2.07839319178881540484981670390, 2.31512589412499989767793667148, 2.36133750393549667424038340583, 2.38256583256639514564599558328, 2.38860418366126324414697801934, 2.41493247290483231253852739880, 2.65684915931880029178794074640, 2.94625236814399587919780433212, 2.96856290998462054614309181260, 3.26254766096715141907415696032, 3.28124396318783369939309500597, 3.30956944547911110256672801830, 3.43425060383822840895468536265, 3.69903853925837043506586809372
Plot not available for L-functions of degree greater than 10.
