| L(s) = 1 | − 9·3-s + 13·4-s + 27·5-s + 38·7-s + 14·8-s + 27·9-s + 19·11-s − 117·12-s − 78·13-s − 243·15-s + 104·16-s − 25·17-s + 169·19-s + 351·20-s − 342·21-s − 28·23-s − 126·24-s + 519·25-s + 54·27-s + 494·28-s + 548·29-s − 118·31-s + 273·32-s − 171·33-s + 1.02e3·35-s + 351·36-s − 151·37-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 13/8·4-s + 2.41·5-s + 2.05·7-s + 0.618·8-s + 9-s + 0.520·11-s − 2.81·12-s − 1.66·13-s − 4.18·15-s + 13/8·16-s − 0.356·17-s + 2.04·19-s + 3.92·20-s − 3.55·21-s − 0.253·23-s − 1.07·24-s + 4.15·25-s + 0.384·27-s + 3.33·28-s + 3.50·29-s − 0.683·31-s + 1.50·32-s − 0.902·33-s + 4.95·35-s + 13/8·36-s − 0.670·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(38.06648485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(38.06648485\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 - 38 T + 2 p^{2} T^{2} + 199 p^{2} T^{3} + 2 p^{5} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | \( ( 1 + p T )^{6} \) |
| good | 2 | \( 1 - 13 T^{2} - 7 p T^{3} + 65 T^{4} + 91 T^{5} - 279 T^{6} + 91 p^{3} T^{7} + 65 p^{6} T^{8} - 7 p^{10} T^{9} - 13 p^{12} T^{10} + p^{18} T^{12} \) |
| 5 | \( 1 - 27 T + 42 p T^{2} - 243 p T^{3} + 31713 T^{4} - 239706 T^{5} - 207299 T^{6} - 239706 p^{3} T^{7} + 31713 p^{6} T^{8} - 243 p^{10} T^{9} + 42 p^{13} T^{10} - 27 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 19 T - 2994 T^{2} + 41545 T^{3} + 5790265 T^{4} - 3690404 p T^{5} - 8133911389 T^{6} - 3690404 p^{4} T^{7} + 5790265 p^{6} T^{8} + 41545 p^{9} T^{9} - 2994 p^{12} T^{10} - 19 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( 1 + 25 T - 12446 T^{2} - 10799 p T^{3} + 5897279 p T^{4} + 727992784 T^{5} - 552185070023 T^{6} + 727992784 p^{3} T^{7} + 5897279 p^{7} T^{8} - 10799 p^{10} T^{9} - 12446 p^{12} T^{10} + 25 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 169 T + 1521 T^{2} - 46294 T^{3} + 163461363 T^{4} - 6420046529 T^{5} - 597903262794 T^{6} - 6420046529 p^{3} T^{7} + 163461363 p^{6} T^{8} - 46294 p^{9} T^{9} + 1521 p^{12} T^{10} - 169 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 + 28 T - 30106 T^{2} - 285390 T^{3} + 562249130 T^{4} + 864579562 T^{5} - 7792606820725 T^{6} + 864579562 p^{3} T^{7} + 562249130 p^{6} T^{8} - 285390 p^{9} T^{9} - 30106 p^{12} T^{10} + 28 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 274 T + 65448 T^{2} - 11935583 T^{3} + 65448 p^{3} T^{4} - 274 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 118 T - 53638 T^{2} - 2874818 T^{3} + 1911903396 T^{4} - 3854063900 T^{5} - 66047986866017 T^{6} - 3854063900 p^{3} T^{7} + 1911903396 p^{6} T^{8} - 2874818 p^{9} T^{9} - 53638 p^{12} T^{10} + 118 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 + 151 T - 71432 T^{2} - 4369791 T^{3} + 2702451603 T^{4} - 177758366768 T^{5} - 161888651711003 T^{6} - 177758366768 p^{3} T^{7} + 2702451603 p^{6} T^{8} - 4369791 p^{9} T^{9} - 71432 p^{12} T^{10} + 151 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 66 T + 142702 T^{2} + 3205533 T^{3} + 142702 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 - 598 T + 277896 T^{2} - 95440011 T^{3} + 277896 p^{3} T^{4} - 598 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 - 682 T + 69 p T^{2} - 10498962 T^{3} + 61544364018 T^{4} - 12264623023586 T^{5} - 1743739491166165 T^{6} - 12264623023586 p^{3} T^{7} + 61544364018 p^{6} T^{8} - 10498962 p^{9} T^{9} + 69 p^{13} T^{10} - 682 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 - 467 T - 219766 T^{2} + 63814569 T^{3} + 63884080781 T^{4} - 8203213169438 T^{5} - 8781264255390811 T^{6} - 8203213169438 p^{3} T^{7} + 63884080781 p^{6} T^{8} + 63814569 p^{9} T^{9} - 219766 p^{12} T^{10} - 467 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 - 289 T - 438052 T^{2} + 56556349 T^{3} + 134964711465 T^{4} - 4333412916752 T^{5} - 31325558028422045 T^{6} - 4333412916752 p^{3} T^{7} + 134964711465 p^{6} T^{8} + 56556349 p^{9} T^{9} - 438052 p^{12} T^{10} - 289 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 577 T - 249486 T^{2} + 126848267 T^{3} + 73462946313 T^{4} - 11972210676902 T^{5} - 17188430249986491 T^{6} - 11972210676902 p^{3} T^{7} + 73462946313 p^{6} T^{8} + 126848267 p^{9} T^{9} - 249486 p^{12} T^{10} - 577 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 839 T - 112074 T^{2} - 313800705 T^{3} + 1589331895 T^{4} + 71431041846086 T^{5} + 39685998745531195 T^{6} + 71431041846086 p^{3} T^{7} + 1589331895 p^{6} T^{8} - 313800705 p^{9} T^{9} - 112074 p^{12} T^{10} + 839 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 1039 T + 1009259 T^{2} - 559869309 T^{3} + 1009259 p^{3} T^{4} - 1039 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 434 T - 799748 T^{2} + 260747994 T^{3} + 442320287838 T^{4} - 72720094585562 T^{5} - 171696182436251705 T^{6} - 72720094585562 p^{3} T^{7} + 442320287838 p^{6} T^{8} + 260747994 p^{9} T^{9} - 799748 p^{12} T^{10} - 434 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 1260 T - 381916 T^{2} + 115210254 T^{3} + 1271960814312 T^{4} - 416468714253792 T^{5} - 372874319828164097 T^{6} - 416468714253792 p^{3} T^{7} + 1271960814312 p^{6} T^{8} + 115210254 p^{9} T^{9} - 381916 p^{12} T^{10} - 1260 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 836 T + 1168832 T^{2} - 997730567 T^{3} + 1168832 p^{3} T^{4} - 836 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 787 T - 948904 T^{2} - 258238959 T^{3} + 766609483757 T^{4} - 178764648441866 T^{5} - 784092763242560767 T^{6} - 178764648441866 p^{3} T^{7} + 766609483757 p^{6} T^{8} - 258238959 p^{9} T^{9} - 948904 p^{12} T^{10} + 787 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 - 267 T + 13485 p T^{2} + 223489775 T^{3} + 13485 p^{4} T^{4} - 267 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−5.98794638786098212238198197888, −5.73258529517903271208602781765, −5.64355671804193258020397831410, −5.48079902433891453377581924083, −5.27444162056172661495755676576, −5.23013722160706710491930158154, −5.02729836551228304951942489616, −4.75615463238603816950215642914, −4.71709379236326167605465277693, −4.34568484487294809658336054929, −4.28849353960563309860821166330, −3.82377742425709016171828819845, −3.61257587665629741505833914976, −3.36682692387648757585059143916, −2.70440791664186003051443262104, −2.68562614112447221814925451327, −2.42600028038690359419539745941, −2.32921773197100663825721697200, −2.27377019989157438151020806778, −1.91810956550769787853106213462, −1.43383820591970581646914326675, −1.09022206722294800858893695483, −0.998369375918931177820236516061, −0.76672972816991804587646911068, −0.72152933332919884732644667902,
0.72152933332919884732644667902, 0.76672972816991804587646911068, 0.998369375918931177820236516061, 1.09022206722294800858893695483, 1.43383820591970581646914326675, 1.91810956550769787853106213462, 2.27377019989157438151020806778, 2.32921773197100663825721697200, 2.42600028038690359419539745941, 2.68562614112447221814925451327, 2.70440791664186003051443262104, 3.36682692387648757585059143916, 3.61257587665629741505833914976, 3.82377742425709016171828819845, 4.28849353960563309860821166330, 4.34568484487294809658336054929, 4.71709379236326167605465277693, 4.75615463238603816950215642914, 5.02729836551228304951942489616, 5.23013722160706710491930158154, 5.27444162056172661495755676576, 5.48079902433891453377581924083, 5.64355671804193258020397831410, 5.73258529517903271208602781765, 5.98794638786098212238198197888
Plot not available for L-functions of degree greater than 10.
