| L(s) = 1 | + 3·3-s − 5-s + 14·7-s − 2·9-s − 33·11-s + 28·13-s − 3·15-s − 5·17-s + 63·19-s + 42·21-s − 33·23-s + 22·25-s − 15·27-s − 100·29-s − 69·31-s − 99·33-s − 14·35-s + 15·37-s + 84·39-s + 124·41-s + 2·45-s + 171·47-s + 37·49-s − 15·51-s − 97·53-s + 33·55-s + 189·57-s + ⋯ |
| L(s) = 1 | + 3-s − 1/5·5-s + 2·7-s − 2/9·9-s − 3·11-s + 2.15·13-s − 1/5·15-s − 0.294·17-s + 3.31·19-s + 2·21-s − 1.43·23-s + 0.879·25-s − 5/9·27-s − 3.44·29-s − 2.22·31-s − 3·33-s − 2/5·35-s + 0.405·37-s + 2.15·39-s + 3.02·41-s + 2/45·45-s + 3.63·47-s + 0.755·49-s − 0.294·51-s − 1.83·53-s + 3/5·55-s + 3.31·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.369625317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.369625317\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2 p T + 159 T^{2} - 172 p T^{3} + 159 p^{2} T^{4} - 2 p^{5} T^{5} + p^{6} T^{6} \) |
| good | 3 | \( 1 - p T + 11 T^{2} - 8 p T^{3} + 73 T^{4} - 5 p^{4} T^{5} + 118 p^{2} T^{6} - 5 p^{6} T^{7} + 73 p^{4} T^{8} - 8 p^{7} T^{9} + 11 p^{8} T^{10} - p^{11} T^{11} + p^{12} T^{12} \) |
| 5 | \( 1 + T - 21 T^{2} - 84 T^{3} - 119 T^{4} + 691 T^{5} + 21334 T^{6} + 691 p^{2} T^{7} - 119 p^{4} T^{8} - 84 p^{6} T^{9} - 21 p^{8} T^{10} + p^{10} T^{11} + p^{12} T^{12} \) |
| 11 | \( 1 + 3 p T + 647 T^{2} + 852 p T^{3} + 99785 T^{4} + 897939 T^{5} + 8784494 T^{6} + 897939 p^{2} T^{7} + 99785 p^{4} T^{8} + 852 p^{7} T^{9} + 647 p^{8} T^{10} + 3 p^{11} T^{11} + p^{12} T^{12} \) |
| 13 | \( ( 1 - 14 T + 239 T^{2} - 308 p T^{3} + 239 p^{2} T^{4} - 14 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 + 5 T - 597 T^{2} - 4140 T^{3} + 189001 T^{4} + 874775 T^{5} - 52481738 T^{6} + 874775 p^{2} T^{7} + 189001 p^{4} T^{8} - 4140 p^{6} T^{9} - 597 p^{8} T^{10} + 5 p^{10} T^{11} + p^{12} T^{12} \) |
| 19 | \( 1 - 63 T + 2687 T^{2} - 85932 T^{3} + 2344025 T^{4} - 55178949 T^{5} + 1125045854 T^{6} - 55178949 p^{2} T^{7} + 2344025 p^{4} T^{8} - 85932 p^{6} T^{9} + 2687 p^{8} T^{10} - 63 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( 1 + 33 T + 1091 T^{2} + 24024 T^{3} + 239153 T^{4} - 2995065 T^{5} - 92805178 T^{6} - 2995065 p^{2} T^{7} + 239153 p^{4} T^{8} + 24024 p^{6} T^{9} + 1091 p^{8} T^{10} + 33 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( ( 1 + 50 T + 107 p T^{2} + 85660 T^{3} + 107 p^{3} T^{4} + 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 31 | \( 1 + 69 T + 4719 T^{2} + 216108 T^{3} + 9775473 T^{4} + 341517135 T^{5} + 11766553198 T^{6} + 341517135 p^{2} T^{7} + 9775473 p^{4} T^{8} + 216108 p^{6} T^{9} + 4719 p^{8} T^{10} + 69 p^{10} T^{11} + p^{12} T^{12} \) |
| 37 | \( 1 - 15 T - 1557 T^{2} - 33940 T^{3} + 790041 T^{4} + 51498435 T^{5} + 375470262 T^{6} + 51498435 p^{2} T^{7} + 790041 p^{4} T^{8} - 33940 p^{6} T^{9} - 1557 p^{8} T^{10} - 15 p^{10} T^{11} + p^{12} T^{12} \) |
| 41 | \( ( 1 - 62 T + 4951 T^{2} - 208228 T^{3} + 4951 p^{2} T^{4} - 62 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( 1 - 3670 T^{2} + 12561503 T^{4} - 23327513140 T^{6} + 12561503 p^{4} T^{8} - 3670 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( 1 - 171 T + 17823 T^{2} - 1380996 T^{3} + 90217905 T^{4} - 5234761233 T^{5} + 263209480846 T^{6} - 5234761233 p^{2} T^{7} + 90217905 p^{4} T^{8} - 1380996 p^{6} T^{9} + 17823 p^{8} T^{10} - 171 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 + 97 T + 1179 T^{2} - 23940 T^{3} + 4735705 T^{4} - 335970173 T^{5} - 51150763946 T^{6} - 335970173 p^{2} T^{7} + 4735705 p^{4} T^{8} - 23940 p^{6} T^{9} + 1179 p^{8} T^{10} + 97 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 - 27 T + 5867 T^{2} - 151848 T^{3} + 13078745 T^{4} - 314287821 T^{5} + 27132604934 T^{6} - 314287821 p^{2} T^{7} + 13078745 p^{4} T^{8} - 151848 p^{6} T^{9} + 5867 p^{8} T^{10} - 27 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 + 89 T - 3749 T^{2} - 138740 T^{3} + 40371865 T^{4} + 372749339 T^{5} - 166841025514 T^{6} + 372749339 p^{2} T^{7} + 40371865 p^{4} T^{8} - 138740 p^{6} T^{9} - 3749 p^{8} T^{10} + 89 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 + 309 T + 53283 T^{2} + 6629904 T^{3} + 664897785 T^{4} + 56444370027 T^{5} + 4088196283606 T^{6} + 56444370027 p^{2} T^{7} + 664897785 p^{4} T^{8} + 6629904 p^{6} T^{9} + 53283 p^{8} T^{10} + 309 p^{10} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 - 12262 T^{2} + 65975471 T^{4} - 264226747348 T^{6} + 65975471 p^{4} T^{8} - 12262 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( 1 - 123 T - 2541 T^{2} + 161020 T^{3} + 74498025 T^{4} - 953468913 T^{5} - 428445487386 T^{6} - 953468913 p^{2} T^{7} + 74498025 p^{4} T^{8} + 161020 p^{6} T^{9} - 2541 p^{8} T^{10} - 123 p^{10} T^{11} + p^{12} T^{12} \) |
| 79 | \( 1 + 201 T + 34179 T^{2} + 4163112 T^{3} + 467823873 T^{4} + 44136675375 T^{5} + 3764890310758 T^{6} + 44136675375 p^{2} T^{7} + 467823873 p^{4} T^{8} + 4163112 p^{6} T^{9} + 34179 p^{8} T^{10} + 201 p^{10} T^{11} + p^{12} T^{12} \) |
| 83 | \( 1 - 24694 T^{2} + 317726975 T^{4} - 2615092039540 T^{6} + 317726975 p^{4} T^{8} - 24694 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 - 91 T - 8109 T^{2} + 1152060 T^{3} + 18525385 T^{4} - 3745251601 T^{5} + 93648370726 T^{6} - 3745251601 p^{2} T^{7} + 18525385 p^{4} T^{8} + 1152060 p^{6} T^{9} - 8109 p^{8} T^{10} - 91 p^{10} T^{11} + p^{12} T^{12} \) |
| 97 | \( ( 1 - 62 T + 17975 T^{2} - 1362020 T^{3} + 17975 p^{2} T^{4} - 62 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−7.57455195786164339841507527283, −7.52146307434365221749030241139, −7.28259512102583409655079770619, −7.10303831273750381029352290831, −6.50795807778883642094099498097, −6.07831321437548390360598137325, −5.90013962078543132379511680334, −5.85077130772826564192465668486, −5.74711746346751441425582878341, −5.45928648422606504367693176980, −5.13709635094292113261776935765, −5.03216062348860277729174690267, −4.94205885273392263948760755102, −4.26414670534588958799845514680, −4.21326002154154364060839411252, −3.85028868833435709603061669761, −3.77497599267691515283010776858, −3.15779391832884547799785363832, −3.06164394083853148528957710539, −2.83330522036510359245155646602, −2.47110605305406459348682624295, −2.01805472905708573708709424587, −1.55376146737051692488260398981, −1.45317394699426049714300474576, −0.48738742295057291694454308674,
0.48738742295057291694454308674, 1.45317394699426049714300474576, 1.55376146737051692488260398981, 2.01805472905708573708709424587, 2.47110605305406459348682624295, 2.83330522036510359245155646602, 3.06164394083853148528957710539, 3.15779391832884547799785363832, 3.77497599267691515283010776858, 3.85028868833435709603061669761, 4.21326002154154364060839411252, 4.26414670534588958799845514680, 4.94205885273392263948760755102, 5.03216062348860277729174690267, 5.13709635094292113261776935765, 5.45928648422606504367693176980, 5.74711746346751441425582878341, 5.85077130772826564192465668486, 5.90013962078543132379511680334, 6.07831321437548390360598137325, 6.50795807778883642094099498097, 7.10303831273750381029352290831, 7.28259512102583409655079770619, 7.52146307434365221749030241139, 7.57455195786164339841507527283
Plot not available for L-functions of degree greater than 10.
