| L(s) = 1 | + 4·4-s − 24·7-s + 56·13-s + 4·16-s + 52·19-s − 62·25-s − 96·28-s − 28·31-s + 4·37-s − 176·43-s + 238·49-s + 224·52-s + 152·61-s − 16·64-s + 180·67-s − 264·73-s + 208·76-s + 392·79-s − 1.34e3·91-s + 48·97-s − 248·100-s + 244·103-s + 552·109-s − 96·112-s − 236·121-s − 112·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 3.42·7-s + 4.30·13-s + 1/4·16-s + 2.73·19-s − 2.47·25-s − 3.42·28-s − 0.903·31-s + 4/37·37-s − 4.09·43-s + 34/7·49-s + 4.30·52-s + 2.49·61-s − 1/4·64-s + 2.68·67-s − 3.61·73-s + 2.73·76-s + 4.96·79-s − 14.7·91-s + 0.494·97-s − 2.47·100-s + 2.36·103-s + 5.06·109-s − 6/7·112-s − 1.95·121-s − 0.903·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4776448327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4776448327\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 12 T + 97 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 5 | \( 1 + 62 T^{2} + 334 p T^{4} + 57288 T^{6} + 1933151 T^{8} + 57288 p^{4} T^{10} + 334 p^{9} T^{12} + 62 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 236 T^{2} + 27290 T^{4} - 206736 T^{6} - 222903181 T^{8} - 206736 p^{4} T^{10} + 27290 p^{8} T^{12} + 236 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 14 T + 350 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( 1 + 182 T^{2} - 96874 T^{4} - 6742008 T^{6} + 7157172959 T^{8} - 6742008 p^{4} T^{10} - 96874 p^{8} T^{12} + 182 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 13 T - 192 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 1550 T^{2} + 1261766 T^{4} + 900630600 T^{6} + 550645070975 T^{8} + 900630600 p^{4} T^{10} + 1261766 p^{8} T^{12} + 1550 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 2150 T^{2} + 2302862 T^{4} - 2150 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 14 T - 1183 T^{2} - 7602 T^{3} + 789764 T^{4} - 7602 p^{2} T^{5} - 1183 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + 262 T^{2} + 5992 T^{3} - 1813073 T^{4} + 5992 p^{2} T^{5} + 262 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6310 T^{2} + 15578574 T^{4} - 6310 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 44 T + 1185 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( 1 + 5398 T^{2} + 13416118 T^{4} + 32187863752 T^{6} + 81192874034911 T^{8} + 32187863752 p^{4} T^{10} + 13416118 p^{8} T^{12} + 5398 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 + 2182 T^{2} - 6141194 T^{4} - 10645201208 T^{6} + 19033288901119 T^{8} - 10645201208 p^{4} T^{10} - 6141194 p^{8} T^{12} + 2182 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 + 4868 T^{2} - 1450966 T^{4} + 4447735824 T^{6} + 198460840609715 T^{8} + 4447735824 p^{4} T^{10} - 1450966 p^{8} T^{12} + 4868 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 76 T - 1297 T^{2} + 28044 T^{3} + 14688992 T^{4} + 28044 p^{2} T^{5} - 1297 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 90 T - 1090 T^{2} - 19080 T^{3} + 27944079 T^{4} - 19080 p^{2} T^{5} - 1090 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3116 T^{2} + 50749526 T^{4} - 3116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 132 T + 2447 T^{2} + 570108 T^{3} + 103059792 T^{4} + 570108 p^{2} T^{5} + 2447 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 196 T + 16922 T^{2} - 1766352 T^{3} + 183287699 T^{4} - 1766352 p^{2} T^{5} + 16922 p^{4} T^{6} - 196 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 5788 T^{2} + 29940710 T^{4} + 5788 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 12766 T^{2} + 39184102 T^{4} - 21674472248 T^{6} + 1161396453909439 T^{8} - 21674472248 p^{4} T^{10} + 39184102 p^{8} T^{12} + 12766 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 12 T + 18817 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.89312900786708503216838322494, −4.65989878105538747419940356099, −4.45826103228241430539705827680, −4.28825778963463645463998202920, −3.92959311159191346596310904198, −3.83813283233777663330516004608, −3.79211423479965401667675703031, −3.66095132006223950084358246157, −3.39191598781618672069825421429, −3.35414572877843118815927755152, −3.32105701881949198540935663928, −3.25653057356438277636495558937, −3.10449272106525102221202156867, −3.07881078667561725164283439829, −2.43355821740430202116638724201, −2.35161684558998687704362868121, −2.07955701358562133481423551080, −1.98802159630345080740322983678, −1.93375269824777193735110132298, −1.50690492784854959641075636471, −1.20761683556591696753573769594, −1.00429830386208752243826456768, −0.882932000678729013994040214464, −0.55569301181051783765251402220, −0.07134758396239701447450004684,
0.07134758396239701447450004684, 0.55569301181051783765251402220, 0.882932000678729013994040214464, 1.00429830386208752243826456768, 1.20761683556591696753573769594, 1.50690492784854959641075636471, 1.93375269824777193735110132298, 1.98802159630345080740322983678, 2.07955701358562133481423551080, 2.35161684558998687704362868121, 2.43355821740430202116638724201, 3.07881078667561725164283439829, 3.10449272106525102221202156867, 3.25653057356438277636495558937, 3.32105701881949198540935663928, 3.35414572877843118815927755152, 3.39191598781618672069825421429, 3.66095132006223950084358246157, 3.79211423479965401667675703031, 3.83813283233777663330516004608, 3.92959311159191346596310904198, 4.28825778963463645463998202920, 4.45826103228241430539705827680, 4.65989878105538747419940356099, 4.89312900786708503216838322494
Plot not available for L-functions of degree greater than 10.
