| L(s) = 1 | + 3·2-s + 2·3-s + 10·4-s + 15·5-s + 6·6-s + 27·8-s + 2·9-s + 45·10-s + 74·11-s + 20·12-s − 88·13-s + 30·15-s + 107·16-s − 52·17-s + 6·18-s + 168·19-s + 150·20-s + 222·22-s + 124·23-s + 54·24-s + 75·25-s − 264·26-s − 76·27-s + 664·29-s + 90·30-s + 320·31-s + 282·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.384·3-s + 5/4·4-s + 1.34·5-s + 0.408·6-s + 1.19·8-s + 2/27·9-s + 1.42·10-s + 2.02·11-s + 0.481·12-s − 1.87·13-s + 0.516·15-s + 1.67·16-s − 0.741·17-s + 0.0785·18-s + 2.02·19-s + 1.67·20-s + 2.15·22-s + 1.12·23-s + 0.459·24-s + 3/5·25-s − 1.99·26-s − 0.541·27-s + 4.25·29-s + 0.547·30-s + 1.85·31-s + 1.55·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(43.10770584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(43.10770584\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 T - T^{2} + 3 p T^{3} - 17 p T^{4} + 27 p^{2} T^{5} - 7 p^{2} T^{6} + 27 p^{5} T^{7} - 17 p^{7} T^{8} + 3 p^{10} T^{9} - p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 - 2 T + 2 T^{2} + 76 T^{3} - 130 T^{4} - 1322 T^{5} + 37942 T^{6} - 1322 p^{3} T^{7} - 130 p^{6} T^{8} + 76 p^{9} T^{9} + 2 p^{12} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 74 T - 94 T^{2} - 2820 T^{3} + 8464910 T^{4} - 173939954 T^{5} - 4385412154 T^{6} - 173939954 p^{3} T^{7} + 8464910 p^{6} T^{8} - 2820 p^{9} T^{9} - 94 p^{12} T^{10} - 74 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( ( 1 + 44 T + 3100 T^{2} + 238206 T^{3} + 3100 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 + 52 T - 288 T^{2} - 752908 T^{3} - 29368424 T^{4} + 63852308 p T^{5} + 334265704502 T^{6} + 63852308 p^{4} T^{7} - 29368424 p^{6} T^{8} - 752908 p^{9} T^{9} - 288 p^{12} T^{10} + 52 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 168 T + 2027 T^{2} + 265592 T^{3} + 114595794 T^{4} - 8656085848 T^{5} + 54832885619 T^{6} - 8656085848 p^{3} T^{7} + 114595794 p^{6} T^{8} + 265592 p^{9} T^{9} + 2027 p^{12} T^{10} - 168 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 - 124 T - 22857 T^{2} + 1105204 T^{3} + 634123426 T^{4} - 14748478492 T^{5} - 7914946666297 T^{6} - 14748478492 p^{3} T^{7} + 634123426 p^{6} T^{8} + 1105204 p^{9} T^{9} - 22857 p^{12} T^{10} - 124 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 332 T + 80572 T^{2} - 13628846 T^{3} + 80572 p^{3} T^{4} - 332 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 - 320 T - 10941 T^{2} + 1763008 T^{3} + 3384500070 T^{4} - 278314179904 T^{5} - 59091058894233 T^{6} - 278314179904 p^{3} T^{7} + 3384500070 p^{6} T^{8} + 1763008 p^{9} T^{9} - 10941 p^{12} T^{10} - 320 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 - 54 T - 136927 T^{2} + 3440998 T^{3} + 12175799130 T^{4} - 142148225534 T^{5} - 707922647296687 T^{6} - 142148225534 p^{3} T^{7} + 12175799130 p^{6} T^{8} + 3440998 p^{9} T^{9} - 136927 p^{12} T^{10} - 54 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 362 T + 248299 T^{2} + 51434996 T^{3} + 248299 p^{3} T^{4} + 362 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 16 T + 149005 T^{2} + 1019664 T^{3} + 149005 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 730 T + 2242 p T^{2} - 1007004 T^{3} + 24109589618 T^{4} + 9993166261882 T^{5} + 2025463443599150 T^{6} + 9993166261882 p^{3} T^{7} + 24109589618 p^{6} T^{8} - 1007004 p^{9} T^{9} + 2242 p^{13} T^{10} + 730 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 + 110 T - 93859 T^{2} + 2392194 p T^{3} + 1784645198 T^{4} - 7131960840034 T^{5} + 7973625047201945 T^{6} - 7131960840034 p^{3} T^{7} + 1784645198 p^{6} T^{8} + 2392194 p^{10} T^{9} - 93859 p^{12} T^{10} + 110 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 + 180 T + 28823 T^{2} + 257689380 T^{3} + 22412251046 T^{4} + 6868299496980 T^{5} + 32474814375635987 T^{6} + 6868299496980 p^{3} T^{7} + 22412251046 p^{6} T^{8} + 257689380 p^{9} T^{9} + 28823 p^{12} T^{10} + 180 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 1222 T + 389525 T^{2} - 162522786 T^{3} + 242940203470 T^{4} - 105335390933270 T^{5} + 21735987281908033 T^{6} - 105335390933270 p^{3} T^{7} + 242940203470 p^{6} T^{8} - 162522786 p^{9} T^{9} + 389525 p^{12} T^{10} - 1222 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 204 T - 50433 T^{2} + 423243644 T^{3} + 22043581110 T^{4} - 11457272075796 T^{5} + 88998297429919803 T^{6} - 11457272075796 p^{3} T^{7} + 22043581110 p^{6} T^{8} + 423243644 p^{9} T^{9} - 50433 p^{12} T^{10} + 204 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 136 T + 900677 T^{2} + 112926832 T^{3} + 900677 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 310 T - 354179 T^{2} + 245357358 T^{3} - 32081304342 T^{4} - 24976640455726 T^{5} + 61371740848252405 T^{6} - 24976640455726 p^{3} T^{7} - 32081304342 p^{6} T^{8} + 245357358 p^{9} T^{9} - 354179 p^{12} T^{10} - 310 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 1034 T - 25106 T^{2} + 220511196 T^{3} - 41077197342 T^{4} + 124653064678486 T^{5} - 153603517219068050 T^{6} + 124653064678486 p^{3} T^{7} - 41077197342 p^{6} T^{8} + 220511196 p^{9} T^{9} - 25106 p^{12} T^{10} - 1034 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 20 p T + 2204369 T^{2} - 1855605736 T^{3} + 2204369 p^{3} T^{4} - 20 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 - 242 T - 1368663 T^{2} + 349740098 T^{3} + 946715505694 T^{4} - 129345951063602 T^{5} - 643596887060304931 T^{6} - 129345951063602 p^{3} T^{7} + 946715505694 p^{6} T^{8} + 349740098 p^{9} T^{9} - 1368663 p^{12} T^{10} - 242 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 + 100 T + 2266056 T^{2} + 184512618 T^{3} + 2266056 p^{3} T^{4} + 100 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−6.16091492065578320301074935681, −6.06090864458501627632005378758, −5.95286352353971532054373597350, −5.30868886972934559752198103496, −5.28500543051983447970429532477, −5.15183712638408995675062558531, −4.96146435182429819971706328369, −4.74077984614148202440302322093, −4.65804935437790212473491903355, −4.48546832456463471730888451208, −4.34098943021363942926588584540, −3.75782099282428544176642939275, −3.43356899703332902171075458616, −3.34768403932301816516682257228, −3.25121492270930399134484257750, −3.23467737825115845404692038690, −2.68486371112356589260739315791, −2.30195453952128192658545431213, −2.23967133125582646153772626910, −2.18428083326076772685647773245, −1.70523714186116733536538597872, −1.19393596140488274514900662994, −1.02151398203568652331894194233, −1.00970099627373616097101907612, −0.45065545964587772571251341883,
0.45065545964587772571251341883, 1.00970099627373616097101907612, 1.02151398203568652331894194233, 1.19393596140488274514900662994, 1.70523714186116733536538597872, 2.18428083326076772685647773245, 2.23967133125582646153772626910, 2.30195453952128192658545431213, 2.68486371112356589260739315791, 3.23467737825115845404692038690, 3.25121492270930399134484257750, 3.34768403932301816516682257228, 3.43356899703332902171075458616, 3.75782099282428544176642939275, 4.34098943021363942926588584540, 4.48546832456463471730888451208, 4.65804935437790212473491903355, 4.74077984614148202440302322093, 4.96146435182429819971706328369, 5.15183712638408995675062558531, 5.28500543051983447970429532477, 5.30868886972934559752198103496, 5.95286352353971532054373597350, 6.06090864458501627632005378758, 6.16091492065578320301074935681
Plot not available for L-functions of degree greater than 10.
