| L(s) = 1 | + 12·11-s − 114·25-s − 210·49-s + 264·59-s − 276·73-s + 396·83-s − 396·97-s + 1.23e3·107-s − 462·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 852·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.09·11-s − 4.55·25-s − 4.28·49-s + 4.47·59-s − 3.78·73-s + 4.77·83-s − 4.08·97-s + 11.5·107-s − 3.81·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.04·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6181866822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6181866822\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 57 T^{2} + 1002 T^{4} + 9857 T^{6} + 1002 p^{4} T^{8} + 57 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 7 | \( ( 1 + 15 p T^{2} + 8826 T^{4} + 474833 T^{6} + 8826 p^{4} T^{8} + 15 p^{9} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 - 3 T + 138 T^{2} - 483 T^{3} + 138 p^{2} T^{4} - 3 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 13 | \( ( 1 - 426 T^{2} + 124239 T^{4} - 25072940 T^{6} + 124239 p^{4} T^{8} - 426 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 - 546 T^{2} + 242367 T^{4} - 90439868 T^{6} + 242367 p^{4} T^{8} - 546 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 19 | \( ( 1 - 306 T^{2} + 257391 T^{4} - 77582428 T^{6} + 257391 p^{4} T^{8} - 306 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 23 | \( ( 1 - 1434 T^{2} + 601023 T^{4} - 128374796 T^{6} + 601023 p^{4} T^{8} - 1434 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 + 2478 T^{2} + 2126127 T^{4} + 1260445412 T^{6} + 2126127 p^{4} T^{8} + 2478 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 + 4617 T^{2} + 9797274 T^{4} + 12047637233 T^{6} + 9797274 p^{4} T^{8} + 4617 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 37 | \( ( 1 - 7530 T^{2} + 24415215 T^{4} - 43755336236 T^{6} + 24415215 p^{4} T^{8} - 7530 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 7746 T^{2} + 28041567 T^{4} - 59757402620 T^{6} + 28041567 p^{4} T^{8} - 7746 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 - 1638 T^{2} + 8806815 T^{4} - 11447892628 T^{6} + 8806815 p^{4} T^{8} - 1638 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 - 2406 T^{2} + 8752719 T^{4} - 25981450580 T^{6} + 8752719 p^{4} T^{8} - 2406 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 + 4017 T^{2} + 12453402 T^{4} + 50692791593 T^{6} + 12453402 p^{4} T^{8} + 4017 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 66 T + 9543 T^{2} - 407004 T^{3} + 9543 p^{2} T^{4} - 66 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 61 | \( ( 1 - 6918 T^{2} + 21319935 T^{4} - 63973116884 T^{6} + 21319935 p^{4} T^{8} - 6918 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 67 | \( ( 1 - 16710 T^{2} + 151257855 T^{4} - 833047801684 T^{6} + 151257855 p^{4} T^{8} - 16710 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 11514 T^{2} + 87698559 T^{4} - 538762144844 T^{6} + 87698559 p^{4} T^{8} - 11514 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 + 69 T + 15618 T^{2} + 733669 T^{3} + 15618 p^{2} T^{4} + 69 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 79 | \( ( 1 + 33006 T^{2} + 474646143 T^{4} + 3839533081508 T^{6} + 474646143 p^{4} T^{8} + 33006 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 83 | \( ( 1 - 99 T + 6330 T^{2} + 20349 T^{3} + 6330 p^{2} T^{4} - 99 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 89 | \( ( 1 - 8070 T^{2} + 80203695 T^{4} - 322161826196 T^{6} + 80203695 p^{4} T^{8} - 8070 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 + 99 T + 12582 T^{2} + 453575 T^{3} + 12582 p^{2} T^{4} + 99 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.69689336369473995072093153351, −2.60657051522855383190446121104, −2.33831039173896838342733559401, −2.22798262820290052660190342489, −2.22459037373444500252825174511, −2.16802037302832292042138053484, −2.06031555436571486944038800524, −2.01518220198001312007553521524, −1.87363262419832595829328620571, −1.86713152290127601335912686592, −1.86084229739213613359377810496, −1.69094138635176081018447224657, −1.61248752741126687493382048614, −1.40224154079959418111567882657, −1.35636302717180701318905344389, −1.29664738438863520573773053562, −1.04409106303624737561412387602, −0.879653954104533514376895772608, −0.77355368633971625700164765701, −0.63904650626269700289834124216, −0.56335359198558670248481576780, −0.44972391133373244047104504665, −0.42895227081085034884019817234, −0.37071587942355454235664701862, −0.02178518514695568688705650196,
0.02178518514695568688705650196, 0.37071587942355454235664701862, 0.42895227081085034884019817234, 0.44972391133373244047104504665, 0.56335359198558670248481576780, 0.63904650626269700289834124216, 0.77355368633971625700164765701, 0.879653954104533514376895772608, 1.04409106303624737561412387602, 1.29664738438863520573773053562, 1.35636302717180701318905344389, 1.40224154079959418111567882657, 1.61248752741126687493382048614, 1.69094138635176081018447224657, 1.86084229739213613359377810496, 1.86713152290127601335912686592, 1.87363262419832595829328620571, 2.01518220198001312007553521524, 2.06031555436571486944038800524, 2.16802037302832292042138053484, 2.22459037373444500252825174511, 2.22798262820290052660190342489, 2.33831039173896838342733559401, 2.60657051522855383190446121104, 2.69689336369473995072093153351
Plot not available for L-functions of degree greater than 10.
