| L(s) = 1 | − 4·2-s + 4·3-s + 8·4-s − 16·6-s + 8·9-s + 32·12-s − 36·16-s − 4·17-s − 32·18-s − 12·23-s + 8·27-s + 4·31-s + 96·32-s + 16·34-s + 64·36-s + 48·46-s + 16·47-s − 144·48-s − 16·51-s + 8·53-s − 32·54-s − 16·61-s − 16·62-s − 96·64-s − 32·68-s − 48·69-s − 7·81-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 4·4-s − 6.53·6-s + 8/3·9-s + 9.23·12-s − 9·16-s − 0.970·17-s − 7.54·18-s − 2.50·23-s + 1.53·27-s + 0.718·31-s + 16.9·32-s + 2.74·34-s + 32/3·36-s + 7.07·46-s + 2.33·47-s − 20.7·48-s − 2.24·51-s + 1.09·53-s − 4.35·54-s − 2.04·61-s − 2.03·62-s − 12·64-s − 3.88·68-s − 5.77·69-s − 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436550755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436550755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 + p T + p T^{2} )^{4}( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 2 T + 2 T^{2} - 64 T^{3} - 353 T^{4} - 64 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + 6 T + 18 T^{2} - 168 T^{3} - 1033 T^{4} - 168 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )( 1 - 529 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 8 T + 32 T^{2} + 496 T^{3} - 4193 T^{4} + 496 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 8 T^{2} + 392 T^{3} - 3593 T^{4} + 392 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 8711 T^{4} + 55730400 T^{8} - 8711 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 8542 T^{4} + p^{4} T^{8} )( 1 + 9791 T^{4} + p^{4} T^{8} ) \) |
| 79 | \( ( 1 + 109 T^{2} + 5640 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 64 T^{2} - 3825 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−4.67337617088692544196036024277, −4.47546784922258891045712145105, −4.33128829542689080849937271635, −4.33001600641972095253018475536, −4.31655917767988480843085532245, −4.15578074481229049625497700986, −3.78371726270320687342662554831, −3.74210156812899871120782348219, −3.66321057162493646453210492360, −3.46503762032844108718405780733, −3.25229871183858720922471779681, −2.97993613209101044209735503330, −2.81854466197300988019900595806, −2.65910799050043790076027193229, −2.46024004159058986217061371195, −2.20029195779545056747025100039, −2.19965650760220021533538898237, −2.03643403302992216206165889242, −1.99832383011435197432469926491, −1.58880674238501735875086698575, −1.54599430225155699533172763868, −1.25362108856905026747014741763, −1.10370690732497524695037684475, −0.78350872324225690542582429530, −0.25663987346834354596711321596,
0.25663987346834354596711321596, 0.78350872324225690542582429530, 1.10370690732497524695037684475, 1.25362108856905026747014741763, 1.54599430225155699533172763868, 1.58880674238501735875086698575, 1.99832383011435197432469926491, 2.03643403302992216206165889242, 2.19965650760220021533538898237, 2.20029195779545056747025100039, 2.46024004159058986217061371195, 2.65910799050043790076027193229, 2.81854466197300988019900595806, 2.97993613209101044209735503330, 3.25229871183858720922471779681, 3.46503762032844108718405780733, 3.66321057162493646453210492360, 3.74210156812899871120782348219, 3.78371726270320687342662554831, 4.15578074481229049625497700986, 4.31655917767988480843085532245, 4.33001600641972095253018475536, 4.33128829542689080849937271635, 4.47546784922258891045712145105, 4.67337617088692544196036024277
Plot not available for L-functions of degree greater than 10.
