| L(s) = 1 | + 16·4-s − 68·7-s + 520·13-s + 64·16-s + 2.17e3·19-s − 898·25-s − 1.08e3·28-s − 812·31-s − 2.14e3·37-s + 7.09e3·43-s − 326·49-s + 8.32e3·52-s + 1.39e4·61-s − 1.02e3·64-s − 3.22e3·67-s + 1.72e4·73-s + 3.48e4·76-s + 1.15e4·79-s − 3.53e4·91-s − 1.58e3·97-s − 1.43e4·100-s − 1.97e4·103-s − 8.16e4·109-s − 4.35e3·112-s − 4.27e4·121-s − 1.29e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 1.38·7-s + 3.07·13-s + 1/4·16-s + 6.02·19-s − 1.43·25-s − 1.38·28-s − 0.844·31-s − 1.56·37-s + 3.83·43-s − 0.135·49-s + 3.07·52-s + 3.74·61-s − 1/4·64-s − 0.718·67-s + 3.23·73-s + 6.02·76-s + 1.85·79-s − 4.27·91-s − 0.167·97-s − 1.43·100-s − 1.86·103-s − 6.87·109-s − 0.346·112-s − 2.91·121-s − 0.844·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(6.575522159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.575522159\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 898 T^{2} + 6701 p T^{4} - 7499198 T^{6} + 129783411076 T^{8} - 7499198 p^{8} T^{10} + 6701 p^{17} T^{12} + 898 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 + 34 T + 271 p T^{2} - 188462 T^{3} - 7991276 T^{4} - 188462 p^{4} T^{5} + 271 p^{9} T^{6} + 34 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 + 42706 T^{2} + 960129265 T^{4} + 18575205696754 T^{6} + 309033367095588964 T^{8} + 18575205696754 p^{8} T^{10} + 960129265 p^{16} T^{12} + 42706 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 20 p T + 16906 T^{2} + 128560 p T^{3} - 176572685 T^{4} + 128560 p^{5} T^{5} + 16906 p^{8} T^{6} - 20 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 141628 T^{2} + 17877746310 T^{4} - 141628 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 - 544 T + 311298 T^{2} - 544 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( 1 + 328228 T^{2} + 45446519626 T^{4} - 30963345777318512 T^{6} - \)\(10\!\cdots\!01\)\( T^{8} - 30963345777318512 p^{8} T^{10} + 45446519626 p^{16} T^{12} + 328228 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 + 1008316 T^{2} + 548953292362 T^{4} - 537175269535551248 T^{6} - \)\(54\!\cdots\!01\)\( T^{8} - 537175269535551248 p^{8} T^{10} + 548953292362 p^{16} T^{12} + 1008316 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 406 T - 13265 p T^{2} - 516022346 T^{3} - 673285790396 T^{4} - 516022346 p^{4} T^{5} - 13265 p^{9} T^{6} + 406 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 + 536 T + 2677074 T^{2} + 536 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 41 | \( 1 + 2818852 T^{2} + 1736492870986 T^{4} - 27513170223926120048 T^{6} - \)\(88\!\cdots\!01\)\( T^{8} - 27513170223926120048 p^{8} T^{10} + 1736492870986 p^{16} T^{12} + 2818852 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 - 3548 T + 2813578 T^{2} - 10420915952 T^{3} + 40820161793971 T^{4} - 10420915952 p^{4} T^{5} + 2813578 p^{8} T^{6} - 3548 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 + 7863868 T^{2} + 12329374820746 T^{4} + 14850692777280199408 T^{6} + \)\(41\!\cdots\!99\)\( T^{8} + 14850692777280199408 p^{8} T^{10} + 12329374820746 p^{16} T^{12} + 7863868 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 6092770 T^{2} + 34232682711939 T^{4} - 6092770 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 + 33635068 T^{2} + 566474079465034 T^{4} + \)\(91\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!55\)\( T^{8} + \)\(91\!\cdots\!64\)\( p^{8} T^{10} + 566474079465034 p^{16} T^{12} + 33635068 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 6968 T + 12082318 T^{2} - 61172239232 T^{3} + 466147632767971 T^{4} - 61172239232 p^{4} T^{5} + 12082318 p^{8} T^{6} - 6968 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 1612 T - 38143382 T^{2} + 708770608 T^{3} + 1201473282065011 T^{4} + 708770608 p^{4} T^{5} - 38143382 p^{8} T^{6} + 1612 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 32170108 T^{2} + 1192261720639110 T^{4} - 32170108 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4306 T + 59542323 T^{2} - 4306 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 79 | \( ( 1 - 5780 T - 48271574 T^{2} - 21847313360 T^{3} + 3853249456653715 T^{4} - 21847313360 p^{4} T^{5} - 48271574 p^{8} T^{6} - 5780 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 + 142639138 T^{2} + 10803890392902049 T^{4} + \)\(71\!\cdots\!94\)\( T^{6} + \)\(40\!\cdots\!40\)\( T^{8} + \)\(71\!\cdots\!94\)\( p^{8} T^{10} + 10803890392902049 p^{16} T^{12} + 142639138 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 - 23105596 T^{2} - 4781996488035066 T^{4} - 23105596 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 790 T - 131427479 T^{2} - 35555516570 T^{3} + 9544934797744180 T^{4} - 35555516570 p^{4} T^{5} - 131427479 p^{8} T^{6} + 790 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−5.29180640272039384631591597575, −5.25370027661849032485974622860, −5.09162033782135126839110963137, −4.57520644859308063529199201576, −4.24509497766475161121501522134, −4.04487953040051385345755643870, −3.87000727039590833691755597841, −3.86056340351696325139460766232, −3.83149947477402590070818678131, −3.55230092099979530930991273400, −3.40087476749825235598568146609, −3.19067322237215482804078372874, −2.95419096720204473875540848908, −2.74083415713023669232495431777, −2.68391546746254359105236705543, −2.65240013780701725534763301073, −2.05462673618919040749244137684, −1.83230230078375139818470930482, −1.60051164654485055771767164613, −1.54470441306244502259117683295, −0.974010797676846072695731432141, −0.950388831901018798163104622525, −0.872358056580916204294594519015, −0.71689711061633177563703893391, −0.14565334129182781204526143906,
0.14565334129182781204526143906, 0.71689711061633177563703893391, 0.872358056580916204294594519015, 0.950388831901018798163104622525, 0.974010797676846072695731432141, 1.54470441306244502259117683295, 1.60051164654485055771767164613, 1.83230230078375139818470930482, 2.05462673618919040749244137684, 2.65240013780701725534763301073, 2.68391546746254359105236705543, 2.74083415713023669232495431777, 2.95419096720204473875540848908, 3.19067322237215482804078372874, 3.40087476749825235598568146609, 3.55230092099979530930991273400, 3.83149947477402590070818678131, 3.86056340351696325139460766232, 3.87000727039590833691755597841, 4.04487953040051385345755643870, 4.24509497766475161121501522134, 4.57520644859308063529199201576, 5.09162033782135126839110963137, 5.25370027661849032485974622860, 5.29180640272039384631591597575
Plot not available for L-functions of degree greater than 10.
