| L(s) = 1 | − 4·2-s + 13·4-s − 24·8-s + 16·13-s + 45·16-s + 20·25-s − 64·26-s − 64·29-s − 44·32-s − 112·37-s + 16·41-s + 168·49-s − 80·50-s + 208·52-s − 352·53-s + 256·58-s − 176·61-s + 29·64-s − 240·73-s + 448·74-s − 64·82-s − 80·89-s + 432·97-s − 672·98-s + 260·100-s + 224·101-s − 384·104-s + ⋯ |
| L(s) = 1 | − 2·2-s + 13/4·4-s − 3·8-s + 1.23·13-s + 2.81·16-s + 4/5·25-s − 2.46·26-s − 2.20·29-s − 1.37·32-s − 3.02·37-s + 0.390·41-s + 24/7·49-s − 8/5·50-s + 4·52-s − 6.64·53-s + 4.41·58-s − 2.88·61-s + 0.453·64-s − 3.28·73-s + 6.05·74-s − 0.780·82-s − 0.898·89-s + 4.45·97-s − 6.85·98-s + 13/5·100-s + 2.21·101-s − 3.69·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.087525373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087525373\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T + 3 T^{2} - p^{4} T^{3} - 13 p^{2} T^{4} - p^{6} T^{5} + 3 p^{4} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| good | 7 | \( 1 - 24 p T^{2} + 13404 T^{4} - 690840 T^{6} + 31402310 T^{8} - 690840 p^{4} T^{10} + 13404 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T + 204 T^{2} + 1736 T^{3} - 634 T^{4} + 1736 p^{2} T^{5} + 204 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 732 T^{2} - 3840 T^{3} + 247238 T^{4} - 3840 p^{2} T^{5} + 732 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 616 T^{2} + 755676 T^{4} - 531969752 T^{6} + 267063306566 T^{8} - 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 32 T + 1212 T^{2} + 46048 T^{3} + 1958438 T^{4} + 46048 p^{2} T^{5} + 1212 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 56 T + 3948 T^{2} + 174856 T^{3} + 6816518 T^{4} + 174856 p^{2} T^{5} + 3948 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 8 T + 4924 T^{2} - 33080 T^{3} + 10990150 T^{4} - 33080 p^{2} T^{5} + 4924 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 3976 T^{2} + 15992796 T^{4} - 34977997112 T^{6} + 80511749221766 T^{8} - 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} - 3976 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 9640 T^{2} + 45901788 T^{4} - 144992767640 T^{6} + 353863561499078 T^{8} - 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} - 9640 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 176 T + 396 p T^{2} + 1644496 T^{3} + 101651558 T^{4} + 1644496 p^{2} T^{5} + 396 p^{5} T^{6} + 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 88 T + 12348 T^{2} + 708776 T^{3} + 62059430 T^{4} + 708776 p^{2} T^{5} + 12348 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 19848 T^{2} + 189118044 T^{4} - 1228457910840 T^{6} + 6186195617725190 T^{8} - 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} - 19848 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 120 T + 19740 T^{2} + 1592520 T^{3} + 158554502 T^{4} + 1592520 p^{2} T^{5} + 19740 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 18184 T^{2} + 266916444 T^{4} - 2437903506104 T^{6} + 19925145362261510 T^{8} - 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} - 18184 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 40 T + 11100 T^{2} + 192920 T^{3} + 121014662 T^{4} + 192920 p^{2} T^{5} + 11100 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 216 T + 43516 T^{2} - 4942440 T^{3} + 582543750 T^{4} - 4942440 p^{2} T^{5} + 43516 p^{4} T^{6} - 216 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−5.78637033860650186731953295504, −5.65114988786692094302632187678, −5.12639584927953566803378220584, −5.00893913766092648658088574882, −4.78061260824715246557973185895, −4.61976572653188327045386594410, −4.55997107543300771463601370119, −4.54983053900970408592468708474, −4.45020735844203609799023981616, −3.69293612372741852465916949608, −3.66093448115693733894819019188, −3.54782087952661178316993144190, −3.31992461228834141145714139882, −3.28514627631965387733431727639, −3.18612944744639846952569206082, −2.81536793444476321906241511140, −2.51314763976849669746618219432, −2.12631096288277793706635003876, −1.98087852772968817053311207708, −1.75094143280538136637803540220, −1.72291709474541861075746628187, −1.40494839326151203836650290626, −1.09960482851967168882947662812, −0.68319979180481208243229548186, −0.24453724710631501383165316715,
0.24453724710631501383165316715, 0.68319979180481208243229548186, 1.09960482851967168882947662812, 1.40494839326151203836650290626, 1.72291709474541861075746628187, 1.75094143280538136637803540220, 1.98087852772968817053311207708, 2.12631096288277793706635003876, 2.51314763976849669746618219432, 2.81536793444476321906241511140, 3.18612944744639846952569206082, 3.28514627631965387733431727639, 3.31992461228834141145714139882, 3.54782087952661178316993144190, 3.66093448115693733894819019188, 3.69293612372741852465916949608, 4.45020735844203609799023981616, 4.54983053900970408592468708474, 4.55997107543300771463601370119, 4.61976572653188327045386594410, 4.78061260824715246557973185895, 5.00893913766092648658088574882, 5.12639584927953566803378220584, 5.65114988786692094302632187678, 5.78637033860650186731953295504
Plot not available for L-functions of degree greater than 10.
