| L(s) = 1 | + 1.40e3·5-s − 7.44e5·13-s − 4.14e5·17-s − 3.56e7·25-s + 9.03e7·29-s − 3.88e7·37-s + 9.64e7·41-s + 8.12e8·49-s − 5.43e8·53-s + 1.49e9·61-s − 1.04e9·65-s − 5.05e9·73-s − 5.80e8·85-s + 1.19e10·89-s + 9.76e9·97-s + 3.55e9·101-s + 5.35e9·109-s − 4.36e10·113-s + 6.62e10·121-s − 6.28e10·125-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2.00·13-s − 0.292·17-s − 3.64·25-s + 4.40·29-s − 0.560·37-s + 0.832·41-s + 2.87·49-s − 1.30·53-s + 1.76·61-s − 0.898·65-s − 2.43·73-s − 0.130·85-s + 2.14·89-s + 1.13·97-s + 0.338·101-s + 0.348·109-s − 2.37·113-s + 2.55·121-s − 2.05·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(9.914183911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.914183911\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 28 p^{2} T + 3709566 p T^{2} - 28 p^{12} T^{3} + p^{20} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 812265284 T^{2} + 6473067245599014 p^{2} T^{4} - 812265284 p^{20} T^{6} + p^{40} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 66229966916 T^{2} + \)\(23\!\cdots\!46\)\( T^{4} - 66229966916 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 372316 T + 181774301142 T^{2} + 372316 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 207396 T - 955880189818 T^{2} + 207396 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12690598758916 T^{2} + \)\(80\!\cdots\!86\)\( T^{4} - 12690598758916 p^{20} T^{6} + p^{40} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3996866950108 p T^{2} + \)\(55\!\cdots\!86\)\( T^{4} - 3996866950108 p^{21} T^{6} + p^{40} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 45174620 T + 1253861156034582 T^{2} - 45174620 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 32388580019324 p T^{2} + \)\(98\!\cdots\!86\)\( T^{4} - 32388580019324 p^{21} T^{6} + p^{40} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 19429116 T + 3062422413466742 T^{2} + 19429116 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 48237980 T + 26926052966081382 T^{2} - 48237980 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 61972849884588356 T^{2} + \)\(18\!\cdots\!86\)\( T^{4} - 61972849884588356 p^{20} T^{6} + p^{40} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132768703911352964 T^{2} + \)\(96\!\cdots\!46\)\( T^{4} - 132768703911352964 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 271998660 T + 150814507320372278 T^{2} + 271998660 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 205104388501524028 T^{2} - \)\(21\!\cdots\!02\)\( T^{4} + 205104388501524028 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 747261284 T + 1531451317218865686 T^{2} - 747261284 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1117646157337216388 T^{2} + \)\(36\!\cdots\!58\)\( T^{4} - 1117646157337216388 p^{20} T^{6} + p^{40} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 7220438861866622276 T^{2} + \)\(31\!\cdots\!46\)\( T^{4} - 7220438861866622276 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 2525618172 T + 6018553152394822694 T^{2} + 2525618172 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3704740809769020028 T^{2} + \)\(95\!\cdots\!18\)\( T^{4} + 3704740809769020028 p^{20} T^{6} + p^{40} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22797542628095660036 T^{2} + \)\(57\!\cdots\!26\)\( T^{4} - 22797542628095660036 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 5976756156 T + 70589225470438600166 T^{2} - 5976756156 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 4884499172 T + \)\(12\!\cdots\!14\)\( T^{2} - 4884499172 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.07097397927189393010048634239, −6.32281974699661597109727912320, −6.23659552344333918615965081114, −6.21505821636801723563991901385, −5.96418717977439977316222954546, −5.33884091136783789077214071939, −5.27568348663678169198496481063, −5.16436868505997446599042761008, −4.76071218882554238305316908060, −4.36424421726664186813844017153, −4.21683038941262170616394790016, −4.11353700108429503226293026824, −3.78747676809659291951796988013, −3.25694826972699711447124801187, −2.93768516476891416694161592079, −2.72358304954842835943414747210, −2.61114203459082865637638088459, −2.26506982049974897228684017062, −1.82337397551361006749924143685, −1.76797466231433435299150405359, −1.59096368672743588328378171922, −0.807015128478562803904235259646, −0.65919768964966161687916396864, −0.47506092518383987881151233912, −0.40882237778403447394657484666,
0.40882237778403447394657484666, 0.47506092518383987881151233912, 0.65919768964966161687916396864, 0.807015128478562803904235259646, 1.59096368672743588328378171922, 1.76797466231433435299150405359, 1.82337397551361006749924143685, 2.26506982049974897228684017062, 2.61114203459082865637638088459, 2.72358304954842835943414747210, 2.93768516476891416694161592079, 3.25694826972699711447124801187, 3.78747676809659291951796988013, 4.11353700108429503226293026824, 4.21683038941262170616394790016, 4.36424421726664186813844017153, 4.76071218882554238305316908060, 5.16436868505997446599042761008, 5.27568348663678169198496481063, 5.33884091136783789077214071939, 5.96418717977439977316222954546, 6.21505821636801723563991901385, 6.23659552344333918615965081114, 6.32281974699661597109727912320, 7.07097397927189393010048634239
