| L(s) = 1 | + 38·4-s − 288·7-s + 596·16-s + 8.77e3·25-s − 1.09e4·28-s − 1.82e4·31-s − 6.28e4·49-s + 2.93e4·64-s + 1.03e5·73-s + 3.67e4·79-s + 3.96e4·97-s + 3.33e5·100-s + 4.03e5·103-s − 1.71e5·112-s + 9.59e5·121-s − 6.94e5·124-s + ⋯ |
| L(s) = 1 | + 1.18·4-s − 2.22·7-s + 0.582·16-s + 2.80·25-s − 2.63·28-s − 3.41·31-s − 3.73·49-s + 0.895·64-s + 2.27·73-s + 0.662·79-s + 0.428·97-s + 3.33·100-s + 3.74·103-s − 1.29·112-s + 5.95·121-s − 4.05·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.311018069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.311018069\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 19 p T^{2} + 53 p^{4} T^{4} - 19 p^{11} T^{6} + p^{20} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 4388 T^{2} + 14354486 T^{4} - 4388 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 7 | \( ( 1 + 72 T + 28666 T^{2} + 72 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 11 | \( ( 1 - 479916 T^{2} + 109294844566 T^{4} - 479916 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 13 | \( ( 1 - 554676 T^{2} + 329997501238 T^{4} - 554676 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 + 2472772 T^{2} + 4954091386310 T^{4} + 2472772 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 - 183108 p T^{2} + 8817270769462 T^{4} - 183108 p^{11} T^{6} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10264540 T^{2} + 101845500766694 T^{4} + 10264540 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 79496964 T^{2} + 83439649020494 p T^{4} - 79496964 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4568 T + 32747274 T^{2} + 4568 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 37 | \( ( 1 + 4197228 T^{2} + 258935079700894 p T^{4} + 4197228 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 41 | \( ( 1 + 261287716 T^{2} + 34588826790454502 T^{4} + 261287716 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 43 | \( ( 1 + 41798676 T^{2} - 9059419941818 p^{2} T^{4} + 41798676 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 47 | \( ( 1 + 264020924 T^{2} + 41754568496276166 T^{4} + 264020924 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 + 237163932 T^{2} + 139779136720856950 T^{4} + 237163932 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 59 | \( ( 1 - 1128860972 T^{2} + 1154001202781424854 T^{4} - 1128860972 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2998868596 T^{2} + 3639478193471277942 T^{4} - 2998868596 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 - 2893941964 T^{2} + 4399089903504510198 T^{4} - 2893941964 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 + 950627228 T^{2} + 4512036538031499942 T^{4} + 950627228 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 - 25916 T + 3191531606 T^{2} - 25916 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 79 | \( ( 1 - 9192 T + 5491137130 T^{2} - 9192 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 83 | \( ( 1 + 1581974452 T^{2} + 2777144179876347830 T^{4} + 1581974452 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 + 3517020388 T^{2} + 41902376233484430374 T^{4} + 3517020388 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 97 | \( ( 1 - 9924 T - 3940384442 T^{2} - 9924 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
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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
−6.02916151333631161150526124800, −5.76822772875564085877334696635, −5.39229408158858655006014264248, −5.35216803363383833394902939666, −5.02223174632535369203649809301, −4.99244129913631680875026952427, −4.69455000288568144410335148450, −4.54963674805319986686250702436, −4.31452996321305815176981290444, −4.01347834640836954898028582661, −3.46800182319455019700905149068, −3.45863041903055885591867187895, −3.44502051410518155018913224731, −3.17682509420505562559308253413, −3.14354162850627889541514946550, −2.91608877068805791940486962501, −2.38680300583912003461460045805, −2.09004525189311721885218884352, −2.03721015210669575772544232556, −1.88419199754884552216383441559, −1.29520488684120796364843411399, −1.25758525319898251327220319091, −0.59977432886278979576140844511, −0.39535235865045428488922575361, −0.36353542944046707852013859225,
0.36353542944046707852013859225, 0.39535235865045428488922575361, 0.59977432886278979576140844511, 1.25758525319898251327220319091, 1.29520488684120796364843411399, 1.88419199754884552216383441559, 2.03721015210669575772544232556, 2.09004525189311721885218884352, 2.38680300583912003461460045805, 2.91608877068805791940486962501, 3.14354162850627889541514946550, 3.17682509420505562559308253413, 3.44502051410518155018913224731, 3.45863041903055885591867187895, 3.46800182319455019700905149068, 4.01347834640836954898028582661, 4.31452996321305815176981290444, 4.54963674805319986686250702436, 4.69455000288568144410335148450, 4.99244129913631680875026952427, 5.02223174632535369203649809301, 5.35216803363383833394902939666, 5.39229408158858655006014264248, 5.76822772875564085877334696635, 6.02916151333631161150526124800
Plot not available for L-functions of degree greater than 10.
