| L(s) = 1 | − 5·4-s + 154·13-s − 969·16-s − 9.42e3·19-s − 5.61e3·25-s − 2.34e4·31-s + 1.81e4·37-s − 4.36e4·43-s − 770·52-s + 1.61e4·61-s + 2.64e4·64-s − 1.44e5·67-s + 1.00e5·73-s + 4.71e4·76-s − 1.01e5·79-s − 4.33e5·97-s + 2.80e4·100-s − 1.45e5·103-s − 2.43e5·109-s − 6.37e4·121-s + 1.17e5·124-s + 127-s + 131-s + 137-s + 139-s − 9.09e4·148-s + 149-s + ⋯ |
| L(s) = 1 | − 0.156·4-s + 0.252·13-s − 0.946·16-s − 5.98·19-s − 1.79·25-s − 4.37·31-s + 2.18·37-s − 3.60·43-s − 0.0394·52-s + 0.555·61-s + 0.808·64-s − 3.93·67-s + 2.19·73-s + 0.935·76-s − 1.83·79-s − 4.67·97-s + 0.280·100-s − 1.34·103-s − 1.96·109-s − 0.395·121-s + 0.683·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.341·148-s + 3.69e−6·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5 T^{2} + 497 p T^{4} - 2083 p^{3} T^{6} + 497 p^{11} T^{8} + 5 p^{20} T^{10} + p^{30} T^{12} \) |
| 5 | \( 1 + 5612 T^{2} + 33696448 T^{4} + 105800333314 T^{6} + 33696448 p^{10} T^{8} + 5612 p^{20} T^{10} + p^{30} T^{12} \) |
| 11 | \( 1 + 63728 T^{2} + 42247579024 T^{4} + 1529842075323718 T^{6} + 42247579024 p^{10} T^{8} + 63728 p^{20} T^{10} + p^{30} T^{12} \) |
| 13 | \( ( 1 - 77 T + 329903 T^{2} + 2901514 T^{3} + 329903 p^{5} T^{4} - 77 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 17 | \( 1 + 366750 p T^{2} + 18438724931679 T^{4} + 32803889999214179716 T^{6} + 18438724931679 p^{10} T^{8} + 366750 p^{21} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 + 4711 T + 14724701 T^{2} + 1423351594 p T^{3} + 14724701 p^{5} T^{4} + 4711 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 + 28426626 T^{2} + 379254398764527 T^{4} + \)\(30\!\cdots\!04\)\( T^{6} + 379254398764527 p^{10} T^{8} + 28426626 p^{20} T^{10} + p^{30} T^{12} \) |
| 29 | \( 1 + 56877260 T^{2} + 2153523833778304 T^{4} + \)\(50\!\cdots\!14\)\( T^{6} + 2153523833778304 p^{10} T^{8} + 56877260 p^{20} T^{10} + p^{30} T^{12} \) |
| 31 | \( ( 1 + 11711 T + 91157164 T^{2} + 584745089971 T^{3} + 91157164 p^{5} T^{4} + 11711 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( ( 1 - 9091 T + 131490511 T^{2} - 1228462202570 T^{3} + 131490511 p^{5} T^{4} - 9091 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 41 | \( 1 - 32763274 T^{2} + 36122988699522367 T^{4} - \)\(94\!\cdots\!80\)\( T^{6} + 36122988699522367 p^{10} T^{8} - 32763274 p^{20} T^{10} + p^{30} T^{12} \) |
| 43 | \( ( 1 + 21843 T + 434197953 T^{2} + 4778588910994 T^{3} + 434197953 p^{5} T^{4} + 21843 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 47 | \( 1 + 762615162 T^{2} + 305120186364825711 T^{4} + \)\(81\!\cdots\!00\)\( T^{6} + 305120186364825711 p^{10} T^{8} + 762615162 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 + 1842711660 T^{2} + 1522579120817053824 T^{4} + \)\(77\!\cdots\!34\)\( T^{6} + 1522579120817053824 p^{10} T^{8} + 1842711660 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( 1 - 405776592 T^{2} + 1198265548187923920 T^{4} - \)\(27\!\cdots\!62\)\( T^{6} + 1198265548187923920 p^{10} T^{8} - 405776592 p^{20} T^{10} + p^{30} T^{12} \) |
| 61 | \( ( 1 - 8078 T + 2037780407 T^{2} - 8299164526604 T^{3} + 2037780407 p^{5} T^{4} - 8078 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( ( 1 + 72325 T + 5554510253 T^{2} + 202408145385274 T^{3} + 5554510253 p^{5} T^{4} + 72325 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 71 | \( 1 + 7343984298 T^{2} + 25986157270151134623 T^{4} + \)\(57\!\cdots\!08\)\( T^{6} + 25986157270151134623 p^{10} T^{8} + 7343984298 p^{20} T^{10} + p^{30} T^{12} \) |
| 73 | \( ( 1 - 50029 T + 6702442195 T^{2} - 208365222113006 T^{3} + 6702442195 p^{5} T^{4} - 50029 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 79 | \( ( 1 + 50997 T + 5666795124 T^{2} + 202281745014889 T^{3} + 5666795124 p^{5} T^{4} + 50997 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 + 15907224432 T^{2} + \)\(11\!\cdots\!44\)\( T^{4} + \)\(52\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!44\)\( p^{10} T^{8} + 15907224432 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( 1 + 30683976398 T^{2} + \)\(40\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{6} + \)\(40\!\cdots\!03\)\( p^{10} T^{8} + 30683976398 p^{20} T^{10} + p^{30} T^{12} \) |
| 97 | \( ( 1 + 216524 T + 31765510816 T^{2} + 3171927842758234 T^{3} + 31765510816 p^{5} T^{4} + 216524 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
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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
−5.72309790356314968275124045320, −5.60168642915157385254567810946, −5.24665984019617560045272701009, −5.21217200717232340297827561880, −4.79121571384532551887343267772, −4.74831443427493306283934172338, −4.71081578002991365983640148470, −4.15020303750955114707764145218, −4.11357894971416472733508662261, −4.05333898663728625533508842222, −3.99007410028048903009140339274, −3.93497501819272442383813250889, −3.51023884326221022104378153115, −3.31136154676245526004651131976, −3.12506015206902247911006870088, −2.68742590377036251162664939591, −2.50001036296071612056367047407, −2.43072112941586149287358492857, −2.09617376469856044958710674293, −2.06129429855006854656781456622, −1.93880419177032028739929995222, −1.49928819231485505370091588478, −1.44990432710320739122975703853, −1.25508954022680944596212275878, −0.932070621254252462349449736612, 0, 0, 0, 0, 0, 0,
0.932070621254252462349449736612, 1.25508954022680944596212275878, 1.44990432710320739122975703853, 1.49928819231485505370091588478, 1.93880419177032028739929995222, 2.06129429855006854656781456622, 2.09617376469856044958710674293, 2.43072112941586149287358492857, 2.50001036296071612056367047407, 2.68742590377036251162664939591, 3.12506015206902247911006870088, 3.31136154676245526004651131976, 3.51023884326221022104378153115, 3.93497501819272442383813250889, 3.99007410028048903009140339274, 4.05333898663728625533508842222, 4.11357894971416472733508662261, 4.15020303750955114707764145218, 4.71081578002991365983640148470, 4.74831443427493306283934172338, 4.79121571384532551887343267772, 5.21217200717232340297827561880, 5.24665984019617560045272701009, 5.60168642915157385254567810946, 5.72309790356314968275124045320
Plot not available for L-functions of degree greater than 10.
