| L(s) = 1 | − 64·4-s + 320·9-s − 28·11-s + 2.56e3·16-s − 5.90e3·19-s − 1.03e4·29-s − 1.12e4·31-s − 2.04e4·36-s + 1.49e4·41-s + 1.79e3·44-s + 6.73e4·49-s + 2.78e4·59-s + 8.89e3·61-s − 8.19e4·64-s + 2.93e4·71-s + 3.78e5·76-s + 1.95e5·79-s − 6.19e4·81-s + 1.73e5·89-s − 8.96e3·99-s + 1.96e5·101-s + 6.26e5·109-s + 6.59e5·116-s − 3.05e5·121-s + 7.22e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.31·9-s − 0.0697·11-s + 5/2·16-s − 3.75·19-s − 2.27·29-s − 2.10·31-s − 2.63·36-s + 1.38·41-s + 0.139·44-s + 4.00·49-s + 1.04·59-s + 0.306·61-s − 5/2·64-s + 0.689·71-s + 7.50·76-s + 3.51·79-s − 1.04·81-s + 2.32·89-s − 0.0918·99-s + 1.91·101-s + 5.04·109-s + 4.55·116-s − 1.89·121-s + 4.21·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.065960256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.065960256\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 13 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| good | 3 | \( 1 - 320 T^{2} + 54788 p T^{4} - 6651056 p^{2} T^{6} + 153554950 p^{4} T^{8} - 6651056 p^{12} T^{10} + 54788 p^{21} T^{12} - 320 p^{30} T^{14} + p^{40} T^{16} \) |
| 7 | \( 1 - 67372 T^{2} + 2522487124 T^{4} - 65714422044308 T^{6} + 1275713253160956950 T^{8} - 65714422044308 p^{10} T^{10} + 2522487124 p^{20} T^{12} - 67372 p^{30} T^{14} + p^{40} T^{16} \) |
| 11 | \( ( 1 + 14 T + 153236 T^{2} + 12364914 T^{3} + 53326943070 T^{4} + 12364914 p^{5} T^{5} + 153236 p^{10} T^{6} + 14 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 17 | \( 1 - 5360120 T^{2} + 14374941392156 T^{4} - 26818523909497508040 T^{6} + \)\(41\!\cdots\!86\)\( T^{8} - 26818523909497508040 p^{10} T^{10} + 14374941392156 p^{20} T^{12} - 5360120 p^{30} T^{14} + p^{40} T^{16} \) |
| 19 | \( ( 1 + 2954 T + 10560076 T^{2} + 18952664230 T^{3} + 38745995591246 T^{4} + 18952664230 p^{5} T^{5} + 10560076 p^{10} T^{6} + 2954 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 23 | \( 1 - 28992640 T^{2} + 439699889055436 T^{4} - \)\(44\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{8} - \)\(44\!\cdots\!80\)\( p^{10} T^{10} + 439699889055436 p^{20} T^{12} - 28992640 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( ( 1 + 5156 T + 1939284 p T^{2} + 178523720220 T^{3} + 1316553955378726 T^{4} + 178523720220 p^{5} T^{5} + 1939284 p^{11} T^{6} + 5156 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 182 p T + 29276956 T^{2} - 247391390714 T^{3} - 1425438660015970 T^{4} - 247391390714 p^{5} T^{5} + 29276956 p^{10} T^{6} + 182 p^{16} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( 1 - 119288808 T^{2} + 9390047634614876 T^{4} - \)\(41\!\cdots\!56\)\( T^{6} + \)\(29\!\cdots\!46\)\( T^{8} - \)\(41\!\cdots\!56\)\( p^{10} T^{10} + 9390047634614876 p^{20} T^{12} - 119288808 p^{30} T^{14} + p^{40} T^{16} \) |
| 41 | \( ( 1 - 7472 T + 284314524 T^{2} - 1521586246896 T^{3} + 38316619013422246 T^{4} - 1521586246896 p^{5} T^{5} + 284314524 p^{10} T^{6} - 7472 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( 1 - 546375840 T^{2} + 169321724985133964 T^{4} - \)\(37\!\cdots\!64\)\( T^{6} + \)\(62\!\cdots\!90\)\( T^{8} - \)\(37\!\cdots\!64\)\( p^{10} T^{10} + 169321724985133964 p^{20} T^{12} - 546375840 p^{30} T^{14} + p^{40} T^{16} \) |
| 47 | \( 1 - 358555916 T^{2} + 47793564150013460 T^{4} + \)\(84\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!22\)\( T^{8} + \)\(84\!\cdots\!28\)\( p^{10} T^{10} + 47793564150013460 p^{20} T^{12} - 358555916 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( 1 - 2079455160 T^{2} + 2258236808142256636 T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(78\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!20\)\( p^{10} T^{10} + 2258236808142256636 p^{20} T^{12} - 2079455160 p^{30} T^{14} + p^{40} T^{16} \) |
| 59 | \( ( 1 - 13930 T + 237933388 T^{2} - 12189403108742 T^{3} + 968957068757353262 T^{4} - 12189403108742 p^{5} T^{5} + 237933388 p^{10} T^{6} - 13930 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4448 T + 2415989596 T^{2} - 3847258551472 T^{3} + 2721890336133169526 T^{4} - 3847258551472 p^{5} T^{5} + 2415989596 p^{10} T^{6} - 4448 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( 1 - 2626301276 T^{2} + 5451896902386250996 T^{4} - \)\(46\!\cdots\!40\)\( T^{6} + \)\(66\!\cdots\!86\)\( T^{8} - \)\(46\!\cdots\!40\)\( p^{10} T^{10} + 5451896902386250996 p^{20} T^{12} - 2626301276 p^{30} T^{14} + p^{40} T^{16} \) |
| 71 | \( ( 1 - 14650 T + 6633476420 T^{2} - 73771108959390 T^{3} + 17508092848271717838 T^{4} - 73771108959390 p^{5} T^{5} + 6633476420 p^{10} T^{6} - 14650 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 - 7511807176 T^{2} + 28818008256490787836 T^{4} - \)\(83\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!26\)\( T^{8} - \)\(83\!\cdots\!72\)\( p^{10} T^{10} + 28818008256490787836 p^{20} T^{12} - 7511807176 p^{30} T^{14} + p^{40} T^{16} \) |
| 79 | \( ( 1 - 97500 T + 14127339788 T^{2} - 871288559222028 T^{3} + 67798663350998500902 T^{4} - 871288559222028 p^{5} T^{5} + 14127339788 p^{10} T^{6} - 97500 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 27349347964 T^{2} + \)\(34\!\cdots\!16\)\( T^{4} - \)\(25\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(25\!\cdots\!60\)\( p^{10} T^{10} + \)\(34\!\cdots\!16\)\( p^{20} T^{12} - 27349347964 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 - 86952 T + 12742678652 T^{2} - 1259409109924184 T^{3} + 79321184737442885094 T^{4} - 1259409109924184 p^{5} T^{5} + 12742678652 p^{10} T^{6} - 86952 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 97 | \( 1 - 52454903288 T^{2} + \)\(13\!\cdots\!36\)\( T^{4} - \)\(20\!\cdots\!16\)\( T^{6} + \)\(20\!\cdots\!26\)\( T^{8} - \)\(20\!\cdots\!16\)\( p^{10} T^{10} + \)\(13\!\cdots\!36\)\( p^{20} T^{12} - 52454903288 p^{30} T^{14} + p^{40} T^{16} \) |
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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
−3.75521635515367055237868824014, −3.73006045692899585791953547840, −3.64463819684987642079266698398, −3.46770248249224987198329607765, −3.38577007471533442587074956249, −3.04084581906302498288553589516, −2.95878064494856776805708986070, −2.91093919063051446707157238351, −2.46722232128764254840082065991, −2.37680240764869986688835308137, −2.18514501754596057043972320235, −2.06075236728345228193412831912, −2.02997435283652079141945298964, −1.99952288803201718084899402864, −1.78187999523416819463412970965, −1.65137548329628754045287865436, −1.53464619771253358120551687114, −1.06016642676618261082048713764, −0.852586145795413426517346041353, −0.797780036809037314674552247432, −0.789776836922723712420784276828, −0.62302905464582362135799245162, −0.34365902565863123350136847546, −0.28981035190986336392612785049, −0.25912350557770873483899434475,
0.25912350557770873483899434475, 0.28981035190986336392612785049, 0.34365902565863123350136847546, 0.62302905464582362135799245162, 0.789776836922723712420784276828, 0.797780036809037314674552247432, 0.852586145795413426517346041353, 1.06016642676618261082048713764, 1.53464619771253358120551687114, 1.65137548329628754045287865436, 1.78187999523416819463412970965, 1.99952288803201718084899402864, 2.02997435283652079141945298964, 2.06075236728345228193412831912, 2.18514501754596057043972320235, 2.37680240764869986688835308137, 2.46722232128764254840082065991, 2.91093919063051446707157238351, 2.95878064494856776805708986070, 3.04084581906302498288553589516, 3.38577007471533442587074956249, 3.46770248249224987198329607765, 3.64463819684987642079266698398, 3.73006045692899585791953547840, 3.75521635515367055237868824014
Plot not available for L-functions of degree greater than 10.
