| L(s) = 1 | − 156·3-s − 1.27e3·5-s − 1.70e4·7-s + 3.78e3·9-s − 7.39e4·11-s − 8.56e4·13-s + 1.98e5·15-s + 3.74e5·17-s − 4.18e5·19-s + 2.66e6·21-s − 1.02e6·23-s − 4.52e5·25-s + 2.93e5·27-s − 3.07e6·29-s − 9.28e6·31-s + 1.15e7·33-s + 2.16e7·35-s − 1.76e7·37-s + 1.33e7·39-s − 4.72e7·41-s + 6.00e7·43-s − 4.80e6·45-s + 4.08e7·47-s + 1.07e8·49-s − 5.84e7·51-s + 8.07e6·53-s + 9.40e7·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.910·5-s − 2.68·7-s + 0.192·9-s − 1.52·11-s − 0.832·13-s + 1.01·15-s + 1.08·17-s − 0.736·19-s + 2.98·21-s − 0.764·23-s − 0.231·25-s + 0.106·27-s − 0.807·29-s − 1.80·31-s + 1.69·33-s + 2.44·35-s − 1.54·37-s + 0.925·39-s − 2.61·41-s + 2.67·43-s − 0.174·45-s + 1.22·47-s + 2.67·49-s − 1.21·51-s + 0.140·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8998912 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8998912 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1919966969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1919966969\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 52 p T + 2284 p^{2} T^{2} + 86072 p^{3} T^{3} + 2284 p^{11} T^{4} + 52 p^{19} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 1272 T + 2070036 T^{2} + 393966854 p T^{3} + 2070036 p^{9} T^{4} + 1272 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 17058 T + 26151072 p T^{2} + 188573075434 p T^{3} + 26151072 p^{10} T^{4} + 17058 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 73974 T + 2778732765 T^{2} + 44497153312780 T^{3} + 2778732765 p^{9} T^{4} + 73974 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 374976 T + 378452421528 T^{2} - 86535045026636022 T^{3} + 378452421528 p^{9} T^{4} - 374976 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 418338 T + 678538095285 T^{2} + 294307734083078340 T^{3} + 678538095285 p^{9} T^{4} + 418338 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 44616 p T + 4031709543477 T^{2} + 2277298243304136464 T^{3} + 4031709543477 p^{9} T^{4} + 44616 p^{19} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3075834 T + 33419160920259 T^{2} + 58301045097658242844 T^{3} + 33419160920259 p^{9} T^{4} + 3075834 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 9286482 T + 43584325295901 T^{2} + \)\(12\!\cdots\!48\)\( T^{3} + 43584325295901 p^{9} T^{4} + 9286482 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 17647776 T + 422551891884468 T^{2} + \)\(12\!\cdots\!86\)\( p T^{3} + 422551891884468 p^{9} T^{4} + 17647776 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 47257110 T + 1622226515041563 T^{2} + \)\(32\!\cdots\!20\)\( T^{3} + 1622226515041563 p^{9} T^{4} + 47257110 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 60023760 T + 2518841308328124 T^{2} - \)\(63\!\cdots\!60\)\( T^{3} + 2518841308328124 p^{9} T^{4} - 60023760 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 40824726 T + 3473474836314168 T^{2} - \)\(85\!\cdots\!22\)\( T^{3} + 3473474836314168 p^{9} T^{4} - 40824726 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8072046 T + 660804482335551 T^{2} + \)\(30\!\cdots\!36\)\( T^{3} + 660804482335551 p^{9} T^{4} - 8072046 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 138035310 T + 23544326414292237 T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + 23544326414292237 p^{9} T^{4} + 138035310 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 264203886 T + 55185039025738935 T^{2} - \)\(65\!\cdots\!40\)\( T^{3} + 55185039025738935 p^{9} T^{4} - 264203886 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 203167074 T + 11045547237099813 T^{2} + \)\(71\!\cdots\!08\)\( T^{3} + 11045547237099813 p^{9} T^{4} + 203167074 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 123067110 T + 95038593276035448 T^{2} + \)\(13\!\cdots\!70\)\( T^{3} + 95038593276035448 p^{9} T^{4} + 123067110 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 433013250 T + 184996030152367959 T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + 184996030152367959 p^{9} T^{4} + 433013250 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 406418748 T - 6408821024487075 T^{2} - \)\(51\!\cdots\!80\)\( T^{3} - 6408821024487075 p^{9} T^{4} + 406418748 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 105365610 T + 274290745881973209 T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + 274290745881973209 p^{9} T^{4} + 105365610 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1365375798 T + 1535126512833912615 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + 1535126512833912615 p^{9} T^{4} - 1365375798 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 669691662 T + 2079640297453933599 T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + 2079640297453933599 p^{9} T^{4} - 669691662 p^{18} T^{5} + p^{27} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.639022860482256399166046232046, −9.014599652034705629361084276042, −8.826764717393155714352547375814, −8.651752706499655594677478399302, −7.75555946496839817723145013954, −7.68403093382093804276234786997, −7.26498835378399812225095959750, −7.26283262158878131995470701416, −6.54251553511129430174338037919, −6.37323908222377420574557372629, −5.81035648931621803596596458663, −5.60503785771485445661039161250, −5.55065088170916096794549653676, −4.88905290085430168690570227121, −4.52137396070175520911259265194, −3.88697921903096864803397438317, −3.62558568856952617711396239124, −3.31593243590267596311438644265, −3.00758231802271605251057444847, −2.49443165303308038942850740315, −2.02116934769278117157710554701, −1.60173645698491292907165654800, −0.63077238981532028815178317176, −0.29722789914916249986744248397, −0.24415187774155127277323809845,
0.24415187774155127277323809845, 0.29722789914916249986744248397, 0.63077238981532028815178317176, 1.60173645698491292907165654800, 2.02116934769278117157710554701, 2.49443165303308038942850740315, 3.00758231802271605251057444847, 3.31593243590267596311438644265, 3.62558568856952617711396239124, 3.88697921903096864803397438317, 4.52137396070175520911259265194, 4.88905290085430168690570227121, 5.55065088170916096794549653676, 5.60503785771485445661039161250, 5.81035648931621803596596458663, 6.37323908222377420574557372629, 6.54251553511129430174338037919, 7.26283262158878131995470701416, 7.26498835378399812225095959750, 7.68403093382093804276234786997, 7.75555946496839817723145013954, 8.651752706499655594677478399302, 8.826764717393155714352547375814, 9.014599652034705629361084276042, 9.639022860482256399166046232046
