| L(s) = 1 | + 5·2-s + 16·4-s + 15·5-s + 4·7-s + 15·8-s + 75·10-s + 5·11-s − 7·13-s + 20·14-s − 47·16-s − 310·17-s − 100·19-s + 240·20-s + 25·22-s + 285·23-s + 75·25-s − 35·26-s + 64·28-s + 115·29-s + 115·31-s − 240·32-s − 1.55e3·34-s + 60·35-s − 768·37-s − 500·38-s + 225·40-s + 580·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2·4-s + 1.34·5-s + 0.215·7-s + 0.662·8-s + 2.37·10-s + 0.137·11-s − 0.149·13-s + 0.381·14-s − 0.734·16-s − 4.42·17-s − 1.20·19-s + 2.68·20-s + 0.242·22-s + 2.58·23-s + 3/5·25-s − 0.264·26-s + 0.431·28-s + 0.736·29-s + 0.666·31-s − 1.32·32-s − 7.81·34-s + 0.289·35-s − 3.41·37-s − 2.13·38-s + 0.889·40-s + 2.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.12238586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.12238586\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| good | 2 | \( 1 - 5 T + 9 T^{2} + 5 p^{2} T^{3} - 61 p T^{4} + 5 p^{5} T^{5} - 31 p^{2} T^{6} + 5 p^{8} T^{7} - 61 p^{7} T^{8} + 5 p^{11} T^{9} + 9 p^{12} T^{10} - 5 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 4 T - 218 T^{2} - 12044 T^{3} - 470 p T^{4} + 1471136 T^{5} + 90890438 T^{6} + 1471136 p^{3} T^{7} - 470 p^{7} T^{8} - 12044 p^{9} T^{9} - 218 p^{12} T^{10} - 4 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 5 T - 1080 T^{2} - 41425 T^{3} - 159980 T^{4} + 25684495 T^{5} + 3368624582 T^{6} + 25684495 p^{3} T^{7} - 159980 p^{6} T^{8} - 41425 p^{9} T^{9} - 1080 p^{12} T^{10} - 5 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 7 T - 5773 T^{2} - 33612 T^{3} + 20844469 T^{4} + 71194013 T^{5} - 52113757154 T^{6} + 71194013 p^{3} T^{7} + 20844469 p^{6} T^{8} - 33612 p^{9} T^{9} - 5773 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 155 T + 20347 T^{2} + 1564790 T^{3} + 20347 p^{3} T^{4} + 155 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 50 T + 206 p T^{2} + 317888 T^{3} + 206 p^{4} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 285 T + 28524 T^{2} - 2256405 T^{3} + 317400684 T^{4} - 13020224865 T^{5} - 1488512348966 T^{6} - 13020224865 p^{3} T^{7} + 317400684 p^{6} T^{8} - 2256405 p^{9} T^{9} + 28524 p^{12} T^{10} - 285 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 - 115 T - 12534 T^{2} - 4624025 T^{3} + 4511720 p T^{4} + 97492036145 T^{5} + 5655360898904 T^{6} + 97492036145 p^{3} T^{7} + 4511720 p^{7} T^{8} - 4624025 p^{9} T^{9} - 12534 p^{12} T^{10} - 115 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 - 115 T - 46916 T^{2} + 4911037 T^{3} + 1145212360 T^{4} - 45632382451 T^{5} - 31639275386878 T^{6} - 45632382451 p^{3} T^{7} + 1145212360 p^{6} T^{8} + 4911037 p^{9} T^{9} - 46916 p^{12} T^{10} - 115 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 384 T + 84036 T^{2} + 16234306 T^{3} + 84036 p^{3} T^{4} + 384 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 580 T + 39825 T^{2} - 4282220 T^{3} + 18874413850 T^{4} - 3916827775780 T^{5} + 28208543319497 T^{6} - 3916827775780 p^{3} T^{7} + 18874413850 p^{6} T^{8} - 4282220 p^{9} T^{9} + 39825 p^{12} T^{10} - 580 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 797 T + 254168 T^{2} - 60937233 T^{3} + 18159967960 T^{4} - 2996533058677 T^{5} + 178186071529462 T^{6} - 2996533058677 p^{3} T^{7} + 18159967960 p^{6} T^{8} - 60937233 p^{9} T^{9} + 254168 p^{12} T^{10} - 797 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 + 145 T - 41292 T^{2} - 79957855 T^{3} - 9038581352 T^{4} + 2459618174785 T^{5} + 3588415823907962 T^{6} + 2459618174785 p^{3} T^{7} - 9038581352 p^{6} T^{8} - 79957855 p^{9} T^{9} - 41292 p^{12} T^{10} + 145 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 400 T + 388459 T^{2} - 106443280 T^{3} + 388459 p^{3} T^{4} - 400 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 - 380 T - 444345 T^{2} + 78042740 T^{3} + 168369306070 T^{4} - 12964114409180 T^{5} - 37187829488683597 T^{6} - 12964114409180 p^{3} T^{7} + 168369306070 p^{6} T^{8} + 78042740 p^{9} T^{9} - 444345 p^{12} T^{10} - 380 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 152 T - 570364 T^{2} + 37659180 T^{3} + 207780466900 T^{4} - 4822768441984 T^{5} - 53658262585089842 T^{6} - 4822768441984 p^{3} T^{7} + 207780466900 p^{6} T^{8} + 37659180 p^{9} T^{9} - 570364 p^{12} T^{10} - 152 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 2 T - 658286 T^{2} + 40449076 T^{3} + 235395581326 T^{4} - 13491305732662 T^{5} - 75538814997629578 T^{6} - 13491305732662 p^{3} T^{7} + 235395581326 p^{6} T^{8} + 40449076 p^{9} T^{9} - 658286 p^{12} T^{10} + 2 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 40 T + 396361 T^{2} - 187438400 T^{3} + 396361 p^{3} T^{4} + 40 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 980 T + 1431584 T^{2} + 778921274 T^{3} + 1431584 p^{3} T^{4} + 980 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 + 1013 T - 505163 T^{2} - 265360862 T^{3} + 669863306119 T^{4} + 107513622088673 T^{5} - 340902024884391226 T^{6} + 107513622088673 p^{3} T^{7} + 669863306119 p^{6} T^{8} - 265360862 p^{9} T^{9} - 505163 p^{12} T^{10} + 1013 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 - 270 T - 1341033 T^{2} + 66689730 T^{3} + 1130197048818 T^{4} + 31258807306650 T^{5} - 742796527633986737 T^{6} + 31258807306650 p^{3} T^{7} + 1130197048818 p^{6} T^{8} + 66689730 p^{9} T^{9} - 1341033 p^{12} T^{10} - 270 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 1020 T + 2161707 T^{2} + 1313072760 T^{3} + 2161707 p^{3} T^{4} + 1020 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 720 T - 2185056 T^{2} - 627148804 T^{3} + 3914793681408 T^{4} + 525582536940432 T^{5} - 3925878298041416250 T^{6} + 525582536940432 p^{3} T^{7} + 3914793681408 p^{6} T^{8} - 627148804 p^{9} T^{9} - 2185056 p^{12} T^{10} + 720 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−5.66442700022471041518212490021, −5.53069238968657897966893684976, −5.20450705108620515400293316589, −4.99987031855558922792452249623, −4.89367165732304538371634187047, −4.72717062766255060109147330791, −4.61672490505051803695201292219, −4.10824151821342773621331874258, −4.10573871059460405852811956766, −4.09028969771938343144417076991, −4.07426697806893248868047565664, −3.63571766682843098476889788874, −3.09325254621515075778209379249, −2.94554571408885565116054645435, −2.89355350135783222998598041858, −2.66356653340541148682512127676, −2.63838422845495727945446969161, −2.06196495605018071920661415387, −2.04837573381708597743935586071, −1.92925186667533687559979927281, −1.80269988389021487185675593274, −1.04656788045226044229322834463, −0.898097363904004679357912518585, −0.58051506996186386159327720069, −0.20286553645390218391544323662,
0.20286553645390218391544323662, 0.58051506996186386159327720069, 0.898097363904004679357912518585, 1.04656788045226044229322834463, 1.80269988389021487185675593274, 1.92925186667533687559979927281, 2.04837573381708597743935586071, 2.06196495605018071920661415387, 2.63838422845495727945446969161, 2.66356653340541148682512127676, 2.89355350135783222998598041858, 2.94554571408885565116054645435, 3.09325254621515075778209379249, 3.63571766682843098476889788874, 4.07426697806893248868047565664, 4.09028969771938343144417076991, 4.10573871059460405852811956766, 4.10824151821342773621331874258, 4.61672490505051803695201292219, 4.72717062766255060109147330791, 4.89367165732304538371634187047, 4.99987031855558922792452249623, 5.20450705108620515400293316589, 5.53069238968657897966893684976, 5.66442700022471041518212490021
Plot not available for L-functions of degree greater than 10.
