| L(s) = 1 | − 10·3-s − 11·4-s + 69·9-s + 110·12-s + 52·16-s + 140·17-s + 290·23-s + 350·25-s − 90·27-s − 68·29-s − 759·36-s + 910·43-s − 520·48-s − 1.25e3·49-s − 1.40e3·51-s + 2.18e3·53-s − 1.00e3·61-s + 1.02e3·64-s − 1.54e3·68-s − 2.90e3·69-s − 3.50e3·75-s + 960·79-s − 708·81-s + 680·87-s − 3.19e3·92-s − 3.85e3·100-s + 1.24e3·101-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 1.37·4-s + 23/9·9-s + 2.64·12-s + 0.812·16-s + 1.99·17-s + 2.62·23-s + 14/5·25-s − 0.641·27-s − 0.435·29-s − 3.51·36-s + 3.22·43-s − 1.56·48-s − 3.64·49-s − 3.84·51-s + 5.64·53-s − 2.10·61-s + 1.99·64-s − 2.74·68-s − 5.05·69-s − 5.38·75-s + 1.36·79-s − 0.971·81-s + 0.837·87-s − 3.61·92-s − 3.84·100-s + 1.22·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.368193523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.368193523\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 11 T^{2} + 69 T^{4} - 209 p^{2} T^{6} - 571 p^{4} T^{8} - 209 p^{8} T^{10} + 69 p^{12} T^{12} + 11 p^{18} T^{14} + p^{24} T^{16} \) |
| 3 | \( ( 1 + 5 T + p T^{2} - 160 T^{3} - 920 T^{4} - 160 p^{3} T^{5} + p^{7} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 5 | \( ( 1 - 7 p^{2} T^{2} + 24 p^{4} T^{4} - 7 p^{8} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( 1 + 1251 T^{2} + 939409 T^{4} + 488257794 T^{6} + 193403260614 T^{8} + 488257794 p^{6} T^{10} + 939409 p^{12} T^{12} + 1251 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 + 27 p^{2} T^{2} + 4738201 T^{4} + 64583082 p^{2} T^{6} + 13178458656894 T^{8} + 64583082 p^{8} T^{10} + 4738201 p^{12} T^{12} + 27 p^{20} T^{14} + p^{24} T^{16} \) |
| 17 | \( ( 1 - 70 T - 5063 T^{2} - 9590 T^{3} + 51050100 T^{4} - 9590 p^{3} T^{5} - 5063 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 + 17283 T^{2} + 132047161 T^{4} + 3473986698 p^{2} T^{6} + 88303858734 p^{4} T^{8} + 3473986698 p^{8} T^{10} + 132047161 p^{12} T^{12} + 17283 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 - 145 T - 3937 T^{2} - 91060 T^{3} + 219254380 T^{4} - 91060 p^{3} T^{5} - 3937 p^{6} T^{6} - 145 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 34 T - 32611 T^{2} - 510374 T^{3} + 517193284 T^{4} - 510374 p^{3} T^{5} - 32611 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 24364 T^{2} + 1090408486 T^{4} - 24364 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 + 168106 T^{2} + 16360298409 T^{4} + 1137720564368954 T^{6} + 62504602619332285124 T^{8} + 1137720564368954 p^{6} T^{10} + 16360298409 p^{12} T^{12} + 168106 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( 1 + 127562 T^{2} + 3195894201 T^{4} + 456156757619482 T^{6} + 71128722756941398964 T^{8} + 456156757619482 p^{6} T^{10} + 3195894201 p^{12} T^{12} + 127562 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 - 455 T + 36243 T^{2} - 5354440 T^{3} + 6385191800 T^{4} - 5354440 p^{3} T^{5} + 36243 p^{6} T^{6} - 455 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 246892 T^{2} + 36497465574 T^{4} - 246892 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 545 T + 256304 T^{2} - 545 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 59 | \( 1 + 468563 T^{2} + 88716567801 T^{4} + 21775832811533818 T^{6} + \)\(60\!\cdots\!14\)\( T^{8} + 21775832811533818 p^{6} T^{10} + 88716567801 p^{12} T^{12} + 468563 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 + 502 T - 94959 T^{2} - 53713498 T^{3} + 11662829084 T^{4} - 53713498 p^{3} T^{5} - 94959 p^{6} T^{6} + 502 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 + 1019851 T^{2} + 607095984969 T^{4} + 257087418738258794 T^{6} + \)\(85\!\cdots\!94\)\( T^{8} + 257087418738258794 p^{6} T^{10} + 607095984969 p^{12} T^{12} + 1019851 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( 1 + 1329867 T^{2} + 1070964854881 T^{4} + 586977780256389522 T^{6} + \)\(24\!\cdots\!54\)\( T^{8} + 586977780256389522 p^{6} T^{10} + 1070964854881 p^{12} T^{12} + 1329867 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( ( 1 - 1323543 T^{2} + 730101524384 T^{4} - 1323543 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 2168348 T^{2} + 1826441009014 T^{4} - 2168348 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 + 2262943 T^{2} + 2903049187921 T^{4} + 2769614243674447858 T^{6} + \)\(21\!\cdots\!74\)\( T^{8} + 2769614243674447858 p^{6} T^{10} + 2903049187921 p^{12} T^{12} + 2262943 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( 1 + 1694911 T^{2} + 649600292289 T^{4} + 944368808906306114 T^{6} + \)\(18\!\cdots\!74\)\( T^{8} + 944368808906306114 p^{6} T^{10} + 649600292289 p^{12} T^{12} + 1694911 p^{18} T^{14} + p^{24} T^{16} \) |
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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
−5.25317799722920081604531344028, −5.25197695492614323637041861974, −5.18739097949554728277277213083, −4.86600695110030073202686501069, −4.62712156217080825869590955680, −4.56556429445112088735192072320, −4.42053467955433170504012937421, −4.27398906573708969725014337863, −4.27232388456026673565472674511, −3.80378183807799651395671400142, −3.48058862792448028688463929424, −3.42584302246843277986124263308, −3.34851378763466217589935829882, −2.97939678783003426053348075906, −2.89103166605888257262451750419, −2.88822270051944047622947185141, −2.18233842210118724626617078097, −2.09066564144783394977493517517, −1.91661180956621047180821609984, −1.45101341068146270114417406577, −0.942758990816178163586799194724, −0.802982030403776512660976083554, −0.77209761631502330140608579373, −0.69386295931389947724831114925, −0.58806893503262310754640350266,
0.58806893503262310754640350266, 0.69386295931389947724831114925, 0.77209761631502330140608579373, 0.802982030403776512660976083554, 0.942758990816178163586799194724, 1.45101341068146270114417406577, 1.91661180956621047180821609984, 2.09066564144783394977493517517, 2.18233842210118724626617078097, 2.88822270051944047622947185141, 2.89103166605888257262451750419, 2.97939678783003426053348075906, 3.34851378763466217589935829882, 3.42584302246843277986124263308, 3.48058862792448028688463929424, 3.80378183807799651395671400142, 4.27232388456026673565472674511, 4.27398906573708969725014337863, 4.42053467955433170504012937421, 4.56556429445112088735192072320, 4.62712156217080825869590955680, 4.86600695110030073202686501069, 5.18739097949554728277277213083, 5.25197695492614323637041861974, 5.25317799722920081604531344028
Plot not available for L-functions of degree greater than 10.
