| L(s) = 1 | + 6·2-s + 12·4-s + 5·5-s + 8·7-s − 16·8-s + 30·10-s − 29·11-s + 20·13-s + 48·14-s − 144·16-s − 38·17-s + 57·19-s + 60·20-s − 174·22-s − 14·23-s + 267·25-s + 120·26-s + 96·28-s + 362·29-s − 88·31-s − 288·32-s − 228·34-s + 40·35-s + 384·37-s + 342·38-s − 80·40-s − 432·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 0.447·5-s + 0.431·7-s − 0.707·8-s + 0.948·10-s − 0.794·11-s + 0.426·13-s + 0.916·14-s − 9/4·16-s − 0.542·17-s + 0.688·19-s + 0.670·20-s − 1.68·22-s − 0.126·23-s + 2.13·25-s + 0.905·26-s + 0.647·28-s + 2.31·29-s − 0.509·31-s − 1.59·32-s − 1.15·34-s + 0.193·35-s + 1.70·37-s + 1.45·38-s − 0.316·40-s − 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.62162134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.62162134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 8 T - 22 p T^{2} + 139 p^{2} T^{3} - 22 p^{4} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| good | 5 | \( 1 - p T - 242 T^{2} + 1291 T^{3} + 31349 T^{4} - 94946 T^{5} - 3671531 T^{6} - 94946 p^{3} T^{7} + 31349 p^{6} T^{8} + 1291 p^{9} T^{9} - 242 p^{12} T^{10} - p^{16} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 29 T - 2564 T^{2} - 60061 T^{3} + 4759283 T^{4} + 58883294 T^{5} - 5900798285 T^{6} + 58883294 p^{3} T^{7} + 4759283 p^{6} T^{8} - 60061 p^{9} T^{9} - 2564 p^{12} T^{10} + 29 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( ( 1 - 10 T + 6508 T^{2} - 43337 T^{3} + 6508 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 + 38 T - 10148 T^{2} - 214318 T^{3} + 64981502 T^{4} + 384215870 T^{5} - 350665294625 T^{6} + 384215870 p^{3} T^{7} + 64981502 p^{6} T^{8} - 214318 p^{9} T^{9} - 10148 p^{12} T^{10} + 38 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 3 p T - 14838 T^{2} + 254831 T^{3} + 166435821 T^{4} - 166961520 T^{5} - 1338012994557 T^{6} - 166961520 p^{3} T^{7} + 166435821 p^{6} T^{8} + 254831 p^{9} T^{9} - 14838 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 + 14 T - 24188 T^{2} - 1111042 T^{3} + 287732402 T^{4} + 11473339118 T^{5} - 3133578440789 T^{6} + 11473339118 p^{3} T^{7} + 287732402 p^{6} T^{8} - 1111042 p^{9} T^{9} - 24188 p^{12} T^{10} + 14 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 181 T + 44385 T^{2} - 9691225 T^{3} + 44385 p^{3} T^{4} - 181 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 88 T - 19176 T^{2} + 2427606 T^{3} - 74706896 T^{4} - 63836425952 T^{5} + 24418143111607 T^{6} - 63836425952 p^{3} T^{7} - 74706896 p^{6} T^{8} + 2427606 p^{9} T^{9} - 19176 p^{12} T^{10} + 88 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 - 384 T + 6558 T^{2} + 23609678 T^{3} - 2205845910 T^{4} - 979004345454 T^{5} + 385902155257863 T^{6} - 979004345454 p^{3} T^{7} - 2205845910 p^{6} T^{8} + 23609678 p^{9} T^{9} + 6558 p^{12} T^{10} - 384 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 216 T + 4638 p T^{2} + 29937897 T^{3} + 4638 p^{4} T^{4} + 216 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 363 T + 118869 T^{2} + 51110485 T^{3} + 118869 p^{3} T^{4} + 363 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 183 T - 263418 T^{2} - 21765813 T^{3} + 50364860037 T^{4} + 1987643355060 T^{5} - 5827318049035865 T^{6} + 1987643355060 p^{3} T^{7} + 50364860037 p^{6} T^{8} - 21765813 p^{9} T^{9} - 263418 p^{12} T^{10} + 183 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 + 396 T - 171222 T^{2} - 33593058 T^{3} + 26399387130 T^{4} - 3085597337094 T^{5} - 6213949690744577 T^{6} - 3085597337094 p^{3} T^{7} + 26399387130 p^{6} T^{8} - 33593058 p^{9} T^{9} - 171222 p^{12} T^{10} + 396 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 + 427 T - 20846 T^{2} - 184632833 T^{3} - 989648387 p T^{4} + 9945164910982 T^{5} + 19570132793727955 T^{6} + 9945164910982 p^{3} T^{7} - 989648387 p^{7} T^{8} - 184632833 p^{9} T^{9} - 20846 p^{12} T^{10} + 427 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 + 7 p T - 325797 T^{2} - 168225036 T^{3} + 60617986357 T^{4} + 21366910723417 T^{5} - 7516285932983282 T^{6} + 21366910723417 p^{3} T^{7} + 60617986357 p^{6} T^{8} - 168225036 p^{9} T^{9} - 325797 p^{12} T^{10} + 7 p^{16} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 32 T - 507154 T^{2} + 145892682 T^{3} + 107477835646 T^{4} - 39824972320442 T^{5} - 19149017206317005 T^{6} - 39824972320442 p^{3} T^{7} + 107477835646 p^{6} T^{8} + 145892682 p^{9} T^{9} - 507154 p^{12} T^{10} + 32 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 395 T + 772713 T^{2} - 166861721 T^{3} + 772713 p^{3} T^{4} - 395 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 373 T - 1002964 T^{2} + 125934813 T^{3} + 757146856945 T^{4} - 40185482983214 T^{5} - 334578208592440487 T^{6} - 40185482983214 p^{3} T^{7} + 757146856945 p^{6} T^{8} + 125934813 p^{9} T^{9} - 1002964 p^{12} T^{10} - 373 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 1364 T + 318072 T^{2} + 576829470 T^{3} - 209637930452 T^{4} - 324379754397308 T^{5} + 410997275277132367 T^{6} - 324379754397308 p^{3} T^{7} - 209637930452 p^{6} T^{8} + 576829470 p^{9} T^{9} + 318072 p^{12} T^{10} - 1364 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 77 T + 1084713 T^{2} - 260668709 T^{3} + 1084713 p^{3} T^{4} - 77 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 1233 T - 175281 T^{2} - 162125424 T^{3} + 290804803809 T^{4} - 425032882464177 T^{5} - 812862837191430722 T^{6} - 425032882464177 p^{3} T^{7} + 290804803809 p^{6} T^{8} - 162125424 p^{9} T^{9} - 175281 p^{12} T^{10} + 1233 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 - 590 T + 2642891 T^{2} - 1009069940 T^{3} + 2642891 p^{3} T^{4} - 590 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−5.62524869189332348865029871611, −5.26900638207473616054126684834, −5.17060111296557596441528025178, −5.13263028780337831471085681751, −4.87066505132065839866633340472, −4.86946700877978206789755968971, −4.60735521975841952261691748678, −4.50274045329319385299629358993, −4.44091684015361031834302843912, −3.89461580265313361246633090322, −3.75136496311226683974630550937, −3.70605692695588219538554065409, −3.33206420274992405635196959697, −3.10277078042029779704157686832, −3.02894013184303166141345998998, −2.98147374383528687196785681858, −2.51593693752710297534547055889, −2.40379140337805068954399009468, −2.01060783014524403637409514505, −1.87907511358686061399562601995, −1.46400295011780878770269902690, −1.14303503443587104245738110320, −0.930542282298799680159045998668, −0.52858287458013579756842990840, −0.25060583921978394059552692693,
0.25060583921978394059552692693, 0.52858287458013579756842990840, 0.930542282298799680159045998668, 1.14303503443587104245738110320, 1.46400295011780878770269902690, 1.87907511358686061399562601995, 2.01060783014524403637409514505, 2.40379140337805068954399009468, 2.51593693752710297534547055889, 2.98147374383528687196785681858, 3.02894013184303166141345998998, 3.10277078042029779704157686832, 3.33206420274992405635196959697, 3.70605692695588219538554065409, 3.75136496311226683974630550937, 3.89461580265313361246633090322, 4.44091684015361031834302843912, 4.50274045329319385299629358993, 4.60735521975841952261691748678, 4.86946700877978206789755968971, 4.87066505132065839866633340472, 5.13263028780337831471085681751, 5.17060111296557596441528025178, 5.26900638207473616054126684834, 5.62524869189332348865029871611
Plot not available for L-functions of degree greater than 10.
