| L(s) = 1 | + 2·3-s + 4-s − 8·5-s + 8·7-s + 2·8-s + 3·9-s − 4·11-s + 2·12-s − 16·15-s − 16-s + 4·17-s − 8·20-s + 16·21-s + 4·24-s + 20·25-s + 6·27-s + 8·28-s + 8·32-s − 8·33-s − 64·35-s + 3·36-s − 16·40-s − 8·43-s − 4·44-s − 24·45-s − 2·48-s + 36·49-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 3.57·5-s + 3.02·7-s + 0.707·8-s + 9-s − 1.20·11-s + 0.577·12-s − 4.13·15-s − 1/4·16-s + 0.970·17-s − 1.78·20-s + 3.49·21-s + 0.816·24-s + 4·25-s + 1.15·27-s + 1.51·28-s + 1.41·32-s − 1.39·33-s − 10.8·35-s + 1/2·36-s − 2.52·40-s − 1.21·43-s − 0.603·44-s − 3.57·45-s − 0.288·48-s + 36/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.005693773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.005693773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} - p T^{3} + p T^{4} - p^{2} T^{5} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 - 2 T + T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 7 | \( ( 1 - T )^{8} \) |
| good | 11 | \( ( 1 + 2 T + 3 p T^{2} + 50 T^{3} + 500 T^{4} + 50 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 34 T^{2} + 593 T^{4} - 8610 T^{6} + 122100 T^{8} - 8610 p^{2} T^{10} + 593 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 33 T^{2} - 50 T^{3} + 644 T^{4} - 50 p T^{5} + 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 64 T^{2} + 1180 T^{4} + 10560 T^{6} - 634906 T^{8} + 10560 p^{2} T^{10} + 1180 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 88 T^{2} + 4220 T^{4} - 145000 T^{6} + 3793350 T^{8} - 145000 p^{2} T^{10} + 4220 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 630306 p^{2} T^{10} + 12689 p^{4} T^{12} - 162 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 - 56 T^{2} + 2684 T^{4} - 115208 T^{6} + 3486342 T^{8} - 115208 p^{2} T^{10} + 2684 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 160 T^{2} + 12124 T^{4} - 608352 T^{6} + 24287270 T^{8} - 608352 p^{2} T^{10} + 12124 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 224 T^{2} + 24956 T^{4} - 1767712 T^{6} + 86340934 T^{8} - 1767712 p^{2} T^{10} + 24956 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T + 116 T^{2} + 324 T^{3} + 6150 T^{4} + 324 p T^{5} + 116 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 354 T^{2} + 55793 T^{4} - 5112098 T^{6} + 297791076 T^{8} - 5112098 p^{2} T^{10} + 55793 p^{4} T^{12} - 354 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T^{2} + 256 T^{3} + 1046 T^{4} + 256 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 4 T + 96 T^{2} - 444 T^{3} + 2126 T^{4} - 444 p T^{5} + 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 96 T^{2} + 232 T^{3} + 2958 T^{4} + 232 p T^{5} + 96 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 12 T + 144 T^{2} + 300 T^{3} + 3470 T^{4} + 300 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 - 130 T^{2} + 3089 T^{4} - 667586 T^{6} + 117965092 T^{8} - 667586 p^{2} T^{10} + 3089 p^{4} T^{12} - 130 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 232 T^{2} + 25980 T^{4} - 1355288 T^{6} + 66491558 T^{8} - 1355288 p^{2} T^{10} + 25980 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 360 T^{2} + 59996 T^{4} - 6261336 T^{6} + 552707974 T^{8} - 6261336 p^{2} T^{10} + 59996 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 82 T^{2} - 11855 T^{4} - 200178 T^{6} + 251318948 T^{8} - 200178 p^{2} T^{10} - 11855 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.86928610945997956625856290697, −4.79243214934287415579099173264, −4.58402468887878953331724057701, −4.55848756151163637461578221710, −4.40969627050858980169701041735, −4.30990576236128458921911936405, −4.03466984420156668370720869976, −3.99125724867813844576208841508, −3.69056027283988201261252647864, −3.59897479534566530568646732256, −3.43617139215066148435000328783, −3.34467753566569263159818840667, −3.28961342638728631178936007248, −3.22013043251160656977204830163, −2.74167686633502892408289158918, −2.50291805213141537860239152466, −2.29004087071605131451401318328, −2.27356208387525985556943881837, −2.14544638349669842420452634714, −1.75407393188271888618815557873, −1.73184245860803363294287008278, −1.29347478524658942069749144785, −1.16218146294091971219121674886, −0.70420965804479226476391564329, −0.42301038969498999891358237875,
0.42301038969498999891358237875, 0.70420965804479226476391564329, 1.16218146294091971219121674886, 1.29347478524658942069749144785, 1.73184245860803363294287008278, 1.75407393188271888618815557873, 2.14544638349669842420452634714, 2.27356208387525985556943881837, 2.29004087071605131451401318328, 2.50291805213141537860239152466, 2.74167686633502892408289158918, 3.22013043251160656977204830163, 3.28961342638728631178936007248, 3.34467753566569263159818840667, 3.43617139215066148435000328783, 3.59897479534566530568646732256, 3.69056027283988201261252647864, 3.99125724867813844576208841508, 4.03466984420156668370720869976, 4.30990576236128458921911936405, 4.40969627050858980169701041735, 4.55848756151163637461578221710, 4.58402468887878953331724057701, 4.79243214934287415579099173264, 4.86928610945997956625856290697
Plot not available for L-functions of degree greater than 10.
