| L(s) = 1 | + 18·3-s − 4·4-s + 30·5-s + 189·9-s − 72·12-s + 540·15-s − 31·16-s − 120·20-s + 56·23-s + 525·25-s + 1.51e3·27-s + 576·31-s − 756·36-s + 332·37-s + 5.67e3·45-s + 96·47-s − 558·48-s − 802·49-s + 308·53-s + 2.08e3·59-s − 2.16e3·60-s + 248·64-s − 1.16e3·67-s + 1.00e3·69-s + 1.06e3·71-s + 9.45e3·75-s − 930·80-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 1/2·4-s + 2.68·5-s + 7·9-s − 1.73·12-s + 9.29·15-s − 0.484·16-s − 1.34·20-s + 0.507·23-s + 21/5·25-s + 10.7·27-s + 3.33·31-s − 7/2·36-s + 1.47·37-s + 18.7·45-s + 0.297·47-s − 1.67·48-s − 2.33·49-s + 0.798·53-s + 4.58·59-s − 4.64·60-s + 0.484·64-s − 2.12·67-s + 1.75·69-s + 1.77·71-s + 14.5·75-s − 1.29·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(200.1380594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(200.1380594\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - p T )^{6} \) |
| 5 | \( ( 1 - p T )^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T^{2} + 47 T^{4} + p^{6} T^{6} + 47 p^{6} T^{8} + p^{14} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 + 802 T^{2} + 304543 T^{4} + 94343516 T^{6} + 304543 p^{6} T^{8} + 802 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 + 886 p T^{2} + 58019383 T^{4} + 164950568324 T^{6} + 58019383 p^{6} T^{8} + 886 p^{13} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 1346 T^{2} - 18224113 T^{4} + 91405101604 T^{6} - 18224113 p^{6} T^{8} - 1346 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( 1 + 1462 p T^{2} + 381628039 T^{4} + 3260386698524 T^{6} + 381628039 p^{6} T^{8} + 1462 p^{13} T^{10} + p^{18} T^{12} \) |
| 23 | \( ( 1 - 28 T + 9957 T^{2} + 1257080 T^{3} + 9957 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 29 | \( 1 + 107278 T^{2} + 5613031079 T^{4} + 173083909993924 T^{6} + 5613031079 p^{6} T^{8} + 107278 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 288 T + 95373 T^{2} - 16398016 T^{3} + 95373 p^{3} T^{4} - 288 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( ( 1 - 166 T + 134339 T^{2} - 13813924 T^{3} + 134339 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 392550 T^{2} + 65515823631 T^{4} + 5956276963486324 T^{6} + 65515823631 p^{6} T^{8} + 392550 p^{12} T^{10} + p^{18} T^{12} \) |
| 43 | \( 1 + 167898 T^{2} + 17757275463 T^{4} + 1598829373108364 T^{6} + 17757275463 p^{6} T^{8} + 167898 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( ( 1 - 48 T + 187437 T^{2} + 3225696 T^{3} + 187437 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( ( 1 - 154 T + 250803 T^{2} - 59517148 T^{3} + 250803 p^{3} T^{4} - 154 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( ( 1 - 1040 T + 659337 T^{2} - 334284320 T^{3} + 659337 p^{3} T^{4} - 1040 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( 1 + 1115710 T^{2} + 559401461431 T^{4} + 162322226576222084 T^{6} + 559401461431 p^{6} T^{8} + 1115710 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( ( 1 + 584 T + 19793 T^{2} - 187697680 T^{3} + 19793 p^{3} T^{4} + 584 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 71 | \( ( 1 - 532 T + 371829 T^{2} - 482157976 T^{3} + 371829 p^{3} T^{4} - 532 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 + 1476678 T^{2} + 1108133530623 T^{4} + 524326150985219924 T^{6} + 1108133530623 p^{6} T^{8} + 1476678 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( 1 + 649210 T^{2} + 356801976031 T^{4} + 97641164525431436 T^{6} + 356801976031 p^{6} T^{8} + 649210 p^{12} T^{10} + p^{18} T^{12} \) |
| 83 | \( 1 + 2530162 T^{2} + 2916380468855 T^{4} + 2053315560734357020 T^{6} + 2916380468855 p^{6} T^{8} + 2530162 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 342 T + 2024103 T^{2} + 471201876 T^{3} + 2024103 p^{3} T^{4} + 342 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 - 1406 T + 1951919 T^{2} - 1639433444 T^{3} + 1951919 p^{3} T^{4} - 1406 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−4.65373309934304937388015097981, −4.06578856020299078175997031077, −3.97379743618628166860216641732, −3.93732599849672536407076968952, −3.87414121501668631345885518995, −3.75033995049429060257948107802, −3.41739047889294735133890554364, −3.17383396434966958745570282928, −3.09247010436275058565313815379, −2.87830680894750883760537362042, −2.80037734563663105944052343802, −2.55413511531942314166241788734, −2.54611547018043129420590280641, −2.51132424948967309616672570234, −2.19244778740010749458144788440, −2.06026352859916346685841394232, −1.90064964020114025610781533867, −1.61923964591889839547319110050, −1.61779266309966773239668010028, −1.25566548165062588526259291134, −1.18083421874254433486135847690, −0.844883549829202385190923874806, −0.75158168061831881524838967917, −0.64395549302188950099032247497, −0.27604139538996374723181536907,
0.27604139538996374723181536907, 0.64395549302188950099032247497, 0.75158168061831881524838967917, 0.844883549829202385190923874806, 1.18083421874254433486135847690, 1.25566548165062588526259291134, 1.61779266309966773239668010028, 1.61923964591889839547319110050, 1.90064964020114025610781533867, 2.06026352859916346685841394232, 2.19244778740010749458144788440, 2.51132424948967309616672570234, 2.54611547018043129420590280641, 2.55413511531942314166241788734, 2.80037734563663105944052343802, 2.87830680894750883760537362042, 3.09247010436275058565313815379, 3.17383396434966958745570282928, 3.41739047889294735133890554364, 3.75033995049429060257948107802, 3.87414121501668631345885518995, 3.93732599849672536407076968952, 3.97379743618628166860216641732, 4.06578856020299078175997031077, 4.65373309934304937388015097981
Plot not available for L-functions of degree greater than 10.
