| L(s) = 1 | + 12·3-s − 32·4-s + 148·9-s − 384·12-s − 520·13-s + 640·16-s + 1.33e3·19-s + 664·25-s + 1.21e3·27-s + 4.88e3·31-s − 4.73e3·36-s − 800·37-s − 6.24e3·39-s − 2.81e3·43-s + 7.68e3·48-s + 1.37e3·49-s + 1.66e4·52-s + 1.60e4·57-s + 2.69e3·61-s − 1.02e4·64-s − 6.96e3·67-s + 2.76e4·73-s + 7.96e3·75-s − 4.27e4·76-s − 1.02e3·79-s + 1.37e4·81-s + 5.85e4·93-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 2·4-s + 1.82·9-s − 8/3·12-s − 3.07·13-s + 5/2·16-s + 3.70·19-s + 1.06·25-s + 1.66·27-s + 5.07·31-s − 3.65·36-s − 0.584·37-s − 4.10·39-s − 1.52·43-s + 10/3·48-s + 4/7·49-s + 6.15·52-s + 4.93·57-s + 0.724·61-s − 5/2·64-s − 1.55·67-s + 5.18·73-s + 1.41·75-s − 7.40·76-s − 0.164·79-s + 2.09·81-s + 6.77·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.507437462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.507437462\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 3 | \( 1 - 4 p T - 4 T^{2} + 68 p^{2} T^{3} - 662 p^{2} T^{4} + 68 p^{6} T^{5} - 4 p^{8} T^{6} - 4 p^{13} T^{7} + p^{16} T^{8} \) |
| 7 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| good | 5 | \( 1 - 664 T^{2} + 437924 T^{4} - 88479624 T^{6} + 4286559494 T^{8} - 88479624 p^{8} T^{10} + 437924 p^{16} T^{12} - 664 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 75544 T^{2} + 2921639132 T^{4} - 72856032921000 T^{6} + 1264800259684876358 T^{8} - 72856032921000 p^{8} T^{10} + 2921639132 p^{16} T^{12} - 75544 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 20 p T + 63068 T^{2} + 7956276 T^{3} + 1342851002 T^{4} + 7956276 p^{4} T^{5} + 63068 p^{8} T^{6} + 20 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 71224 T^{2} + 9382405916 T^{4} - 1222841841739656 T^{6} + 85209970415957819462 T^{8} - 1222841841739656 p^{8} T^{10} + 9382405916 p^{16} T^{12} - 71224 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 - 668 T + 562204 T^{2} - 226309484 T^{3} + 108623604730 T^{4} - 226309484 p^{4} T^{5} + 562204 p^{8} T^{6} - 668 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 954952 T^{2} + 314690473244 T^{4} - 8418785376240120 T^{6} - \)\(14\!\cdots\!22\)\( T^{8} - 8418785376240120 p^{8} T^{10} + 314690473244 p^{16} T^{12} - 954952 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 4654456 T^{2} + 10061048290844 T^{4} - 13139168714386018632 T^{6} + \)\(11\!\cdots\!42\)\( T^{8} - 13139168714386018632 p^{8} T^{10} + 10061048290844 p^{16} T^{12} - 4654456 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 - 2440 T + 4750052 T^{2} - 5895563736 T^{3} + 6660667523078 T^{4} - 5895563736 p^{4} T^{5} + 4750052 p^{8} T^{6} - 2440 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 + 400 T + 2084716 T^{2} + 395204848 T^{3} + 5059454317606 T^{4} + 395204848 p^{4} T^{5} + 2084716 p^{8} T^{6} + 400 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 19751576 T^{2} + 177774313337692 T^{4} - \)\(95\!\cdots\!12\)\( T^{6} + \)\(32\!\cdots\!94\)\( T^{8} - \)\(95\!\cdots\!12\)\( p^{8} T^{10} + 177774313337692 p^{16} T^{12} - 19751576 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 1408 T + 10046860 T^{2} + 11972332672 T^{3} + 48969670597606 T^{4} + 11972332672 p^{4} T^{5} + 10046860 p^{8} T^{6} + 1408 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 24321800 T^{2} + 280807668157468 T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(20\!\cdots\!20\)\( p^{8} T^{10} + 280807668157468 p^{16} T^{12} - 24321800 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 46463768 T^{2} + 1024233601864924 T^{4} - \)\(14\!\cdots\!96\)\( T^{6} + \)\(13\!\cdots\!94\)\( T^{8} - \)\(14\!\cdots\!96\)\( p^{8} T^{10} + 1024233601864924 p^{16} T^{12} - 46463768 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 + 10637576 T^{2} + 84111045669860 T^{4} + 49194962777684533080 T^{6} - \)\(15\!\cdots\!78\)\( T^{8} + 49194962777684533080 p^{8} T^{10} + 84111045669860 p^{16} T^{12} + 10637576 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 1348 T + 29646524 T^{2} - 20047475892 T^{3} + 480742537431098 T^{4} - 20047475892 p^{4} T^{5} + 29646524 p^{8} T^{6} - 1348 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 3480 T + 50068124 T^{2} + 145959249576 T^{3} + 1178892510670470 T^{4} + 145959249576 p^{4} T^{5} + 50068124 p^{8} T^{6} + 3480 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 43758904 T^{2} + 1149222526263836 T^{4} - \)\(38\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(38\!\cdots\!08\)\( p^{8} T^{10} + 1149222526263836 p^{16} T^{12} - 43758904 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 13824 T + 129639884 T^{2} - 924716795904 T^{3} + 5270348883746598 T^{4} - 924716795904 p^{4} T^{5} + 129639884 p^{8} T^{6} - 13824 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 512 T + 115971620 T^{2} + 58605823488 T^{3} + 6343534677753926 T^{4} + 58605823488 p^{4} T^{5} + 115971620 p^{8} T^{6} + 512 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 190162744 T^{2} + 15994881716551652 T^{4} - \)\(80\!\cdots\!60\)\( T^{6} + \)\(35\!\cdots\!38\)\( T^{8} - \)\(80\!\cdots\!60\)\( p^{8} T^{10} + 15994881716551652 p^{16} T^{12} - 190162744 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 240564248 T^{2} + 37258790807545948 T^{4} - \)\(36\!\cdots\!72\)\( T^{6} + \)\(27\!\cdots\!86\)\( T^{8} - \)\(36\!\cdots\!72\)\( p^{8} T^{10} + 37258790807545948 p^{16} T^{12} - 240564248 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 13704 T + 113687708 T^{2} + 121903428792 T^{3} - 333706776381114 T^{4} + 121903428792 p^{4} T^{5} + 113687708 p^{8} T^{6} + 13704 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94477191448057933739302706322, −6.94352217699111046347287990663, −6.66671921458207778148184959048, −6.38916709202995514804328157851, −6.24554211601366264253457498554, −5.93125375260446459406446752095, −5.38908138941423016707213879592, −5.22202016127420124349226771956, −5.15900607126687867869351658318, −5.15174021553366816964319172085, −4.68482167815059195726229074543, −4.66688584780610086941732400347, −4.54027947635743140789482835043, −4.20306223735757317565131996215, −3.68580370284497017265600537452, −3.55083575373499602034637554591, −3.48575976348390378866342428201, −2.79383347038542062982775082099, −2.77629435352663285337587270359, −2.68773701569136997878060250287, −2.23491538136301964201733568251, −1.43740858706091604531327737753, −1.19816482167351549059170056366, −0.71359067118230111325631782161, −0.50557307915793631667769541787,
0.50557307915793631667769541787, 0.71359067118230111325631782161, 1.19816482167351549059170056366, 1.43740858706091604531327737753, 2.23491538136301964201733568251, 2.68773701569136997878060250287, 2.77629435352663285337587270359, 2.79383347038542062982775082099, 3.48575976348390378866342428201, 3.55083575373499602034637554591, 3.68580370284497017265600537452, 4.20306223735757317565131996215, 4.54027947635743140789482835043, 4.66688584780610086941732400347, 4.68482167815059195726229074543, 5.15174021553366816964319172085, 5.15900607126687867869351658318, 5.22202016127420124349226771956, 5.38908138941423016707213879592, 5.93125375260446459406446752095, 6.24554211601366264253457498554, 6.38916709202995514804328157851, 6.66671921458207778148184959048, 6.94352217699111046347287990663, 6.94477191448057933739302706322
Plot not available for L-functions of degree greater than 10.
