| L(s) = 1 | − 48·4-s + 530·13-s + 1.53e3·16-s + 836·17-s + 416·23-s + 9.73e3·25-s − 1.87e4·29-s − 2.42e4·43-s + 4.89e4·49-s − 2.54e4·52-s + 4.23e4·53-s − 3.19e3·61-s − 4.09e4·64-s − 4.01e4·68-s − 1.69e5·79-s − 1.99e4·92-s − 4.67e5·100-s − 1.38e5·101-s − 3.64e5·103-s + 5.22e4·107-s + 3.13e5·113-s + 9.01e5·116-s + 4.88e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.869·13-s + 3/2·16-s + 0.701·17-s + 0.163·23-s + 3.11·25-s − 4.14·29-s − 1.99·43-s + 2.90·49-s − 1.30·52-s + 2.07·53-s − 0.109·61-s − 5/4·64-s − 1.05·68-s − 3.05·79-s − 0.245·92-s − 4.67·100-s − 1.35·101-s − 3.38·103-s + 0.440·107-s + 2.31·113-s + 6.22·116-s + 3.03·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.117908104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.117908104\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 530 T - 12785 p T^{2} + 92996 p^{3} T^{3} - 12785 p^{6} T^{4} - 530 p^{10} T^{5} + p^{15} T^{6} \) |
| good | 5 | \( 1 - 9734 T^{2} + 45022279 T^{4} - 151321754836 T^{6} + 45022279 p^{10} T^{8} - 9734 p^{20} T^{10} + p^{30} T^{12} \) |
| 7 | \( 1 - 998 p^{2} T^{2} + 1501179695 T^{4} - 28969294894196 T^{6} + 1501179695 p^{10} T^{8} - 998 p^{22} T^{10} + p^{30} T^{12} \) |
| 11 | \( 1 - 488550 T^{2} + 122692550487 T^{4} - 21659528044972276 T^{6} + 122692550487 p^{10} T^{8} - 488550 p^{20} T^{10} + p^{30} T^{12} \) |
| 17 | \( ( 1 - 418 T + 1867887 T^{2} + 596217860 T^{3} + 1867887 p^{5} T^{4} - 418 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 19 | \( 1 - 4259886 T^{2} + 15059505195735 T^{4} - 52704394717503498500 T^{6} + 15059505195735 p^{10} T^{8} - 4259886 p^{20} T^{10} + p^{30} T^{12} \) |
| 23 | \( ( 1 - 208 T + 13024485 T^{2} - 5279785312 T^{3} + 13024485 p^{5} T^{4} - 208 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 29 | \( ( 1 + 9394 T + 75091011 T^{2} + 387910039372 T^{3} + 75091011 p^{5} T^{4} + 9394 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( 1 - 116335670 T^{2} + 6009258573309407 T^{4} - \)\(20\!\cdots\!76\)\( T^{6} + 6009258573309407 p^{10} T^{8} - 116335670 p^{20} T^{10} + p^{30} T^{12} \) |
| 37 | \( 1 - 289018862 T^{2} + 38604779572315415 T^{4} - \)\(32\!\cdots\!76\)\( T^{6} + 38604779572315415 p^{10} T^{8} - 289018862 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( 1 - 461990702 T^{2} + 96326950481430703 T^{4} - \)\(13\!\cdots\!04\)\( T^{6} + 96326950481430703 p^{10} T^{8} - 461990702 p^{20} T^{10} + p^{30} T^{12} \) |
| 43 | \( ( 1 + 12100 T + 192229169 T^{2} + 42503254792 p T^{3} + 192229169 p^{5} T^{4} + 12100 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 47 | \( 1 - 5196642 p T^{2} + 98471047533657423 T^{4} - \)\(23\!\cdots\!32\)\( T^{6} + 98471047533657423 p^{10} T^{8} - 5196642 p^{21} T^{10} + p^{30} T^{12} \) |
| 53 | \( ( 1 - 21198 T + 82289499 T^{2} + 9230236358604 T^{3} + 82289499 p^{5} T^{4} - 21198 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 59 | \( 1 - 2269058726 T^{2} + 2942781419203692343 T^{4} - \)\(25\!\cdots\!56\)\( T^{6} + 2942781419203692343 p^{10} T^{8} - 2269058726 p^{20} T^{10} + p^{30} T^{12} \) |
| 61 | \( ( 1 + 1598 T + 927987683 T^{2} + 20079713107796 T^{3} + 927987683 p^{5} T^{4} + 1598 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 - 3626333582 T^{2} + 8616065023269477815 T^{4} - \)\(14\!\cdots\!76\)\( T^{6} + 8616065023269477815 p^{10} T^{8} - 3626333582 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( 1 - 5654364942 T^{2} + 254047452086597385 p T^{4} - \)\(37\!\cdots\!88\)\( T^{6} + 254047452086597385 p^{11} T^{8} - 5654364942 p^{20} T^{10} + p^{30} T^{12} \) |
| 73 | \( 1 - 10335109526 T^{2} + 47996850402258820223 T^{4} - \)\(12\!\cdots\!28\)\( T^{6} + 47996850402258820223 p^{10} T^{8} - 10335109526 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 + 84664 T + 9958851181 T^{2} + 485196394773392 T^{3} + 9958851181 p^{5} T^{4} + 84664 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 22485370838 T^{2} + \)\(21\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(21\!\cdots\!95\)\( p^{10} T^{8} - 22485370838 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( 1 - 25940313102 T^{2} + \)\(31\!\cdots\!55\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{6} + \)\(31\!\cdots\!55\)\( p^{10} T^{8} - 25940313102 p^{20} T^{10} + p^{30} T^{12} \) |
| 97 | \( 1 - 34764784422 T^{2} + \)\(54\!\cdots\!75\)\( T^{4} - \)\(54\!\cdots\!16\)\( T^{6} + \)\(54\!\cdots\!75\)\( p^{10} T^{8} - 34764784422 p^{20} T^{10} + p^{30} T^{12} \) |
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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
−5.62665225768673021523670576017, −5.56135087788758856848683297164, −5.36552063976034427609362083327, −5.30125081705098699679772420701, −4.87091060317311492091443754455, −4.73139053089582003258059214155, −4.66864683987543627432447143579, −4.26912031861352398738872011437, −4.01227762182539385907210120643, −3.99004334248554720880488743533, −3.84950229464134239014966395960, −3.47281876708367259058623711348, −3.16468289705507836034764247710, −3.13724758851073491568825200625, −3.10334374090735823617379332998, −2.42126937376574835936121129463, −2.38688013680351612716556171628, −2.06720072838183849428592502977, −1.59883081265575553430319927898, −1.54004811445409200498052538396, −1.11967767587726226799828916492, −1.10631504410509656553720295572, −0.50354109323186260384007720550, −0.43622204193111135896320550062, −0.37921355488310683196211631948,
0.37921355488310683196211631948, 0.43622204193111135896320550062, 0.50354109323186260384007720550, 1.10631504410509656553720295572, 1.11967767587726226799828916492, 1.54004811445409200498052538396, 1.59883081265575553430319927898, 2.06720072838183849428592502977, 2.38688013680351612716556171628, 2.42126937376574835936121129463, 3.10334374090735823617379332998, 3.13724758851073491568825200625, 3.16468289705507836034764247710, 3.47281876708367259058623711348, 3.84950229464134239014966395960, 3.99004334248554720880488743533, 4.01227762182539385907210120643, 4.26912031861352398738872011437, 4.66864683987543627432447143579, 4.73139053089582003258059214155, 4.87091060317311492091443754455, 5.30125081705098699679772420701, 5.36552063976034427609362083327, 5.56135087788758856848683297164, 5.62665225768673021523670576017
Plot not available for L-functions of degree greater than 10.
