L(s) = 1 | + 12·5-s − 8·7-s + 12·9-s + 12·11-s + 24·13-s + 88·19-s + 72·25-s + 24·29-s − 16·31-s − 96·35-s + 32·37-s − 180·41-s + 144·45-s + 36·47-s + 32·49-s − 72·53-s + 144·55-s − 228·59-s − 192·61-s − 96·63-s + 288·65-s + 16·67-s + 36·71-s + 160·73-s − 96·77-s + 48·79-s + 90·81-s + ⋯ |
L(s) = 1 | + 12/5·5-s − 8/7·7-s + 4/3·9-s + 1.09·11-s + 1.84·13-s + 4.63·19-s + 2.87·25-s + 0.827·29-s − 0.516·31-s − 2.74·35-s + 0.864·37-s − 4.39·41-s + 16/5·45-s + 0.765·47-s + 0.653·49-s − 1.35·53-s + 2.61·55-s − 3.86·59-s − 3.14·61-s − 1.52·63-s + 4.43·65-s + 0.238·67-s + 0.507·71-s + 2.19·73-s − 1.24·77-s + 0.607·79-s + 10/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(16.15904600\) |
\(L(\frac12)\) |
\(\approx\) |
\(16.15904600\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 13 | \( 1 - 24 T + 388 T^{2} - 360 p T^{3} + 246 p^{2} T^{4} - 360 p^{3} T^{5} + 388 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
good | 5 | \( 1 - 12 T + 72 T^{2} - 348 T^{3} + 2452 T^{4} - 18612 T^{5} + 107352 T^{6} - 522948 T^{7} + 2524326 T^{8} - 522948 p^{2} T^{9} + 107352 p^{4} T^{10} - 18612 p^{6} T^{11} + 2452 p^{8} T^{12} - 348 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 + 8 T + 32 T^{2} + 88 T^{3} - 1636 T^{4} - 5816 T^{5} + 9696 T^{6} + 84312 T^{7} + 1561414 T^{8} + 84312 p^{2} T^{9} + 9696 p^{4} T^{10} - 5816 p^{6} T^{11} - 1636 p^{8} T^{12} + 88 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 - 12 T + 72 T^{2} + 2148 T^{3} - 140 p^{2} T^{4} - 5364 T^{5} + 3591000 T^{6} - 4652100 T^{7} + 69892134 T^{8} - 4652100 p^{2} T^{9} + 3591000 p^{4} T^{10} - 5364 p^{6} T^{11} - 140 p^{10} T^{12} + 2148 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) |
| 17 | \( 1 - 1016 T^{2} + 594652 T^{4} - 256650440 T^{6} + 84536113606 T^{8} - 256650440 p^{4} T^{10} + 594652 p^{8} T^{12} - 1016 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 88 T + 3872 T^{2} - 114440 T^{3} + 2792348 T^{4} - 66199832 T^{5} + 1561870560 T^{6} - 34451668488 T^{7} + 687884234758 T^{8} - 34451668488 p^{2} T^{9} + 1561870560 p^{4} T^{10} - 66199832 p^{6} T^{11} + 2792348 p^{8} T^{12} - 114440 p^{10} T^{13} + 3872 p^{12} T^{14} - 88 p^{14} T^{15} + p^{16} T^{16} \) |
| 23 | \( 1 - 1352 T^{2} + 1211164 T^{4} - 919326200 T^{6} + 546870337222 T^{8} - 919326200 p^{4} T^{10} + 1211164 p^{8} T^{12} - 1352 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 12 T + 2404 T^{2} - 10740 T^{3} + 2553462 T^{4} - 10740 p^{2} T^{5} + 2404 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 + 16 T + 128 T^{2} + 4400 T^{3} + 1526300 T^{4} + 33831440 T^{5} + 355616640 T^{6} + 22570639536 T^{7} + 1151898602182 T^{8} + 22570639536 p^{2} T^{9} + 355616640 p^{4} T^{10} + 33831440 p^{6} T^{11} + 1526300 p^{8} T^{12} + 4400 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) |
| 37 | \( 1 - 32 T + 512 T^{2} - 113248 T^{3} + 2123708 T^{4} + 18764000 T^{5} + 4724768256 T^{6} - 111804825696 T^{7} - 2252621074874 T^{8} - 111804825696 p^{2} T^{9} + 4724768256 p^{4} T^{10} + 18764000 p^{6} T^{11} + 2123708 p^{8} T^{12} - 113248 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) |
| 41 | \( 1 + 180 T + 16200 T^{2} + 1077252 T^{3} + 65891188 T^{4} + 3775499532 T^{5} + 192388605912 T^{6} + 8753174479068 T^{7} + 369429048471078 T^{8} + 8753174479068 p^{2} T^{9} + 192388605912 p^{4} T^{10} + 3775499532 p^{6} T^{11} + 65891188 p^{8} T^{12} + 1077252 p^{10} T^{13} + 16200 p^{12} T^{14} + 180 p^{14} T^{15} + p^{16} T^{16} \) |
| 43 | \( 1 - 9704 T^{2} + 46942108 T^{4} - 145828988120 T^{6} + 318232746184198 T^{8} - 145828988120 p^{4} T^{10} + 46942108 p^{8} T^{12} - 9704 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 36 T + 648 T^{2} - 203220 T^{3} + 4042708 T^{4} + 382284036 T^{5} + 4267284120 T^{6} + 315919318356 T^{7} - 83839689122202 T^{8} + 315919318356 p^{2} T^{9} + 4267284120 p^{4} T^{10} + 382284036 p^{6} T^{11} + 4042708 p^{8} T^{12} - 203220 p^{10} T^{13} + 648 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 36 T + 6052 T^{2} + 433596 T^{3} + 17242230 T^{4} + 433596 p^{2} T^{5} + 6052 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 228 T + 25992 T^{2} + 2616180 T^{3} + 265981876 T^{4} + 22067956476 T^{5} + 1540292051736 T^{6} + 106557827242188 T^{7} + 6867086075503590 T^{8} + 106557827242188 p^{2} T^{9} + 1540292051736 p^{4} T^{10} + 22067956476 p^{6} T^{11} + 265981876 p^{8} T^{12} + 2616180 p^{10} T^{13} + 25992 p^{12} T^{14} + 228 p^{14} T^{15} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 96 T + 16036 T^{2} + 1016352 T^{3} + 91399206 T^{4} + 1016352 p^{2} T^{5} + 16036 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 16 T + 128 T^{2} - 69488 T^{3} + 74830748 T^{4} - 906063824 T^{5} + 7332976512 T^{6} - 4028918600112 T^{7} + 2213363729350918 T^{8} - 4028918600112 p^{2} T^{9} + 7332976512 p^{4} T^{10} - 906063824 p^{6} T^{11} + 74830748 p^{8} T^{12} - 69488 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) |
| 71 | \( 1 - 36 T + 648 T^{2} - 1156308 T^{3} + 52863796 T^{4} + 1532170692 T^{5} + 579110210712 T^{6} - 32483951612268 T^{7} - 1080005772645210 T^{8} - 32483951612268 p^{2} T^{9} + 579110210712 p^{4} T^{10} + 1532170692 p^{6} T^{11} + 52863796 p^{8} T^{12} - 1156308 p^{10} T^{13} + 648 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) |
| 73 | \( 1 - 160 T + 12800 T^{2} - 1291712 T^{3} + 71099900 T^{4} + 1493854336 T^{5} - 314835468288 T^{6} + 44981288744928 T^{7} - 5001122834269562 T^{8} + 44981288744928 p^{2} T^{9} - 314835468288 p^{4} T^{10} + 1493854336 p^{6} T^{11} + 71099900 p^{8} T^{12} - 1291712 p^{10} T^{13} + 12800 p^{12} T^{14} - 160 p^{14} T^{15} + p^{16} T^{16} \) |
| 79 | \( ( 1 - 24 T + 17668 T^{2} - 588552 T^{3} + 143245062 T^{4} - 588552 p^{2} T^{5} + 17668 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 12 T + 72 T^{2} + 156132 T^{3} - 66148556 T^{4} + 666125388 T^{5} + 8957792088 T^{6} - 2187511634436 T^{7} + 3174741677260134 T^{8} - 2187511634436 p^{2} T^{9} + 8957792088 p^{4} T^{10} + 666125388 p^{6} T^{11} - 66148556 p^{8} T^{12} + 156132 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) |
| 89 | \( 1 - 60 T + 1800 T^{2} + 288660 T^{3} + 2412628 T^{4} - 4107851460 T^{5} + 283790655000 T^{6} + 13498586694540 T^{7} - 6984420788021274 T^{8} + 13498586694540 p^{2} T^{9} + 283790655000 p^{4} T^{10} - 4107851460 p^{6} T^{11} + 2412628 p^{8} T^{12} + 288660 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \) |
| 97 | \( 1 - 416 T + 86528 T^{2} - 11971840 T^{3} + 1263350396 T^{4} - 125003246272 T^{5} + 14348643876864 T^{6} - 1773872116458144 T^{7} + 191787280414737670 T^{8} - 1773872116458144 p^{2} T^{9} + 14348643876864 p^{4} T^{10} - 125003246272 p^{6} T^{11} + 1263350396 p^{8} T^{12} - 11971840 p^{10} T^{13} + 86528 p^{12} T^{14} - 416 p^{14} T^{15} + p^{16} T^{16} \) |
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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
−5.83334997035610206887307519275, −5.54503062709514896125542361305, −5.48238408798751432753977377803, −5.10730603217487490354146809626, −4.95498992024679537797426143501, −4.82502667260433278618287203835, −4.80557303906700220403322897866, −4.69572201147283387329559743568, −4.60430464458041502922492186939, −3.97614764359033392902706789167, −3.93811725606744623950354369659, −3.50369228236151632686274322922, −3.49495439572271045802482207057, −3.30962342154773955384675972168, −3.18791784779207540680097987522, −3.15985133199766105068840014317, −3.05130794560738177349655490434, −2.42918267419924887767220702086, −2.08012702184040378854789811703, −1.87490308967734872209532679582, −1.77334465423322805867314741825, −1.43956879458254923958440913599, −1.25065757863848383196200406871, −0.885136588302033539763289389000, −0.67747638676501171582990430968,
0.67747638676501171582990430968, 0.885136588302033539763289389000, 1.25065757863848383196200406871, 1.43956879458254923958440913599, 1.77334465423322805867314741825, 1.87490308967734872209532679582, 2.08012702184040378854789811703, 2.42918267419924887767220702086, 3.05130794560738177349655490434, 3.15985133199766105068840014317, 3.18791784779207540680097987522, 3.30962342154773955384675972168, 3.49495439572271045802482207057, 3.50369228236151632686274322922, 3.93811725606744623950354369659, 3.97614764359033392902706789167, 4.60430464458041502922492186939, 4.69572201147283387329559743568, 4.80557303906700220403322897866, 4.82502667260433278618287203835, 4.95498992024679537797426143501, 5.10730603217487490354146809626, 5.48238408798751432753977377803, 5.54503062709514896125542361305, 5.83334997035610206887307519275
Plot not available for L-functions of degree greater than 10.