| L(s) = 1 | − 32·4-s + 576·16-s − 256·25-s + 352·37-s − 3.29e3·43-s − 7.68e3·64-s + 1.60e3·67-s − 6.04e3·79-s + 8.19e3·100-s + 5.79e3·109-s + 1.44e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.12e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.06e4·169-s + 1.05e5·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4·4-s + 9·16-s − 2.04·25-s + 1.56·37-s − 11.6·43-s − 15·64-s + 2.91·67-s − 8.61·79-s + 8.19·100-s + 5.08·109-s + 1.08·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 6.25·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.86·169-s + 46.7·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6151553390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6151553390\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 128 T^{2} + 60096 T^{4} + 5594752 T^{6} + 1399856546 T^{8} + 5594752 p^{6} T^{10} + 60096 p^{12} T^{12} + 128 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 11 | \( ( 1 - 720 T^{2} + 1042300 T^{4} - 1278052272 T^{6} + 6076372496166 T^{8} - 1278052272 p^{6} T^{10} + 1042300 p^{12} T^{12} - 720 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 13 | \( ( 1 - 5344 T^{2} + 8811456 T^{4} + 1175160352 T^{6} - 22788372799390 T^{8} + 1175160352 p^{6} T^{10} + 8811456 p^{12} T^{12} - 5344 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 17 | \( ( 1 + 16144 T^{2} + 178577344 T^{4} + 1274112016144 T^{6} + 7323562577141122 T^{8} + 1274112016144 p^{6} T^{10} + 178577344 p^{12} T^{12} + 16144 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 19 | \( ( 1 - 37896 T^{2} + 705115836 T^{4} - 8318582270328 T^{6} + 67823212142716262 T^{8} - 8318582270328 p^{6} T^{10} + 705115836 p^{12} T^{12} - 37896 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 23 | \( ( 1 - 57344 T^{2} + 1709505340 T^{4} - 33477664629248 T^{6} + 474373943001845830 T^{8} - 33477664629248 p^{6} T^{10} + 1709505340 p^{12} T^{12} - 57344 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 29 | \( ( 1 - 14320 T^{2} + 1186196796 T^{4} - 32515591805072 T^{6} + 726114934256020838 T^{8} - 32515591805072 p^{6} T^{10} + 1186196796 p^{12} T^{12} - 14320 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 31 | \( ( 1 - 125032 T^{2} + 8786514588 T^{4} - 420567619845080 T^{6} + 14499987744382327622 T^{8} - 420567619845080 p^{6} T^{10} + 8786514588 p^{12} T^{12} - 125032 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 37 | \( ( 1 - 88 T + 115072 T^{2} - 15701176 T^{3} + 7276730962 T^{4} - 15701176 p^{3} T^{5} + 115072 p^{6} T^{6} - 88 p^{9} T^{7} + p^{12} T^{8} )^{4} \) |
| 41 | \( ( 1 + 373952 T^{2} + 65857198272 T^{4} + 7366845760676032 T^{6} + \)\(58\!\cdots\!98\)\( T^{8} + 7366845760676032 p^{6} T^{10} + 65857198272 p^{12} T^{12} + 373952 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 43 | \( ( 1 + 824 T + 531924 T^{2} + 210724792 T^{3} + 71081985062 T^{4} + 210724792 p^{3} T^{5} + 531924 p^{6} T^{6} + 824 p^{9} T^{7} + p^{12} T^{8} )^{4} \) |
| 47 | \( ( 1 + 357992 T^{2} + 70686275868 T^{4} + 10603118539669720 T^{6} + \)\(12\!\cdots\!78\)\( T^{8} + 10603118539669720 p^{6} T^{10} + 70686275868 p^{12} T^{12} + 357992 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 53 | \( ( 1 - 268032 T^{2} + 83863944484 T^{4} - 14561358758841600 T^{6} + \)\(27\!\cdots\!82\)\( T^{8} - 14561358758841600 p^{6} T^{10} + 83863944484 p^{12} T^{12} - 268032 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 59 | \( ( 1 + 837576 T^{2} + 314131075452 T^{4} + 72461480316899640 T^{6} + \)\(14\!\cdots\!78\)\( T^{8} + 72461480316899640 p^{6} T^{10} + 314131075452 p^{12} T^{12} + 837576 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 61 | \( ( 1 - 758224 T^{2} + 264639647424 T^{4} - 72890841392749136 T^{6} + \)\(18\!\cdots\!74\)\( T^{8} - 72890841392749136 p^{6} T^{10} + 264639647424 p^{12} T^{12} - 758224 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 67 | \( ( 1 - 400 T + 385252 T^{2} - 54557776 T^{3} + 64548428518 T^{4} - 54557776 p^{3} T^{5} + 385252 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8} )^{4} \) |
| 71 | \( ( 1 + 33584 T^{2} + 339666757596 T^{4} + 8018556032558800 T^{6} + \)\(59\!\cdots\!62\)\( T^{8} + 8018556032558800 p^{6} T^{10} + 339666757596 p^{12} T^{12} + 33584 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 73 | \( ( 1 - 41424 T^{2} + 116468956896 T^{4} - 76845121215813840 T^{6} + \)\(34\!\cdots\!78\)\( T^{8} - 76845121215813840 p^{6} T^{10} + 116468956896 p^{12} T^{12} - 41424 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 79 | \( ( 1 + 1512 T + 1384236 T^{2} + 1246654920 T^{3} + 1037327871974 T^{4} + 1246654920 p^{3} T^{5} + 1384236 p^{6} T^{6} + 1512 p^{9} T^{7} + p^{12} T^{8} )^{4} \) |
| 83 | \( ( 1 + 3385560 T^{2} + 5430493263420 T^{4} + 5428638329879379624 T^{6} + \)\(37\!\cdots\!50\)\( T^{8} + 5428638329879379624 p^{6} T^{10} + 5430493263420 p^{12} T^{12} + 3385560 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 89 | \( ( 1 + 3282720 T^{2} + 4630973023296 T^{4} + 3966251307803111712 T^{6} + \)\(27\!\cdots\!06\)\( T^{8} + 3966251307803111712 p^{6} T^{10} + 4630973023296 p^{12} T^{12} + 3282720 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
| 97 | \( ( 1 - 1699056 T^{2} + 603035339040 T^{4} - 590758774593695472 T^{6} + \)\(12\!\cdots\!30\)\( T^{8} - 590758774593695472 p^{6} T^{10} + 603035339040 p^{12} T^{12} - 1699056 p^{18} T^{14} + p^{24} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.06002119292104509109654144598, −2.06000664631291513432701154652, −1.88757079812716850093861977592, −1.82956535885383588727887700478, −1.69489253718502132514621845330, −1.68732332763626941222527269184, −1.60018726594109155153051026007, −1.59054477309771626760311505056, −1.57699326537944327960681780424, −1.55965858997023617869354221926, −1.49118852038453148288923799688, −1.34540877106596354041047838940, −1.20474993511488235910363564996, −1.05593374534093886355289691098, −0.865640781856555824871895225868, −0.853204622038086096862785803070, −0.804162252099698369842665962473, −0.55528460060061496942718204837, −0.50158971762960290515183216115, −0.44749242130165278514560243362, −0.40868828883249409180321641565, −0.33922510088025673449899169420, −0.27413485215289566156917048109, −0.18720906937233954514599434089, −0.05668346461002194545000309182,
0.05668346461002194545000309182, 0.18720906937233954514599434089, 0.27413485215289566156917048109, 0.33922510088025673449899169420, 0.40868828883249409180321641565, 0.44749242130165278514560243362, 0.50158971762960290515183216115, 0.55528460060061496942718204837, 0.804162252099698369842665962473, 0.853204622038086096862785803070, 0.865640781856555824871895225868, 1.05593374534093886355289691098, 1.20474993511488235910363564996, 1.34540877106596354041047838940, 1.49118852038453148288923799688, 1.55965858997023617869354221926, 1.57699326537944327960681780424, 1.59054477309771626760311505056, 1.60018726594109155153051026007, 1.68732332763626941222527269184, 1.69489253718502132514621845330, 1.82956535885383588727887700478, 1.88757079812716850093861977592, 2.06000664631291513432701154652, 2.06002119292104509109654144598
Plot not available for L-functions of degree greater than 10.
