| L(s) = 1 | − 2·3-s + 4-s + 6·7-s + 9-s + 6·11-s − 2·12-s + 4·13-s − 6·19-s − 12·21-s + 6·23-s + 2·27-s + 6·28-s + 6·29-s − 12·33-s + 36-s − 8·39-s + 12·41-s − 2·43-s + 6·44-s + 16·49-s + 4·52-s − 12·53-s + 12·57-s − 48·59-s − 20·61-s + 6·63-s − 64-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 2.26·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 1.10·13-s − 1.37·19-s − 2.61·21-s + 1.25·23-s + 0.384·27-s + 1.13·28-s + 1.11·29-s − 2.08·33-s + 1/6·36-s − 1.28·39-s + 1.87·41-s − 0.304·43-s + 0.904·44-s + 16/7·49-s + 0.554·52-s − 1.64·53-s + 1.58·57-s − 6.24·59-s − 2.56·61-s + 0.755·63-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192977324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192977324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 6 T + 20 T^{2} - 48 T^{3} + 99 T^{4} - 48 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 108 T^{3} - 573 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 133 T^{2} - 1020 T^{3} + 7512 T^{4} - 1020 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 52 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 24 T + 251 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 68 T^{2} - 336 T^{3} - 549 T^{4} - 336 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.21979453775535520421196817304, −6.20201852156915303392570568172, −6.09045037045612896269046654805, −6.07829124670115205132284388890, −6.04428309238695196799159404662, −5.41200375168885157617239214757, −5.22013205456007714625647560862, −4.97833521983088396506763165975, −4.76460831320810299779223629097, −4.49908253404808572890183658174, −4.48375720134840243426692458265, −4.45002159499126000701755496321, −4.09759702845009636101798731704, −3.58857897256405516883179885675, −3.49342218613693284100468809283, −3.10424065668116919733448059319, −3.09279991862754643048718570422, −2.50768481078637230066765402810, −2.38877505130440582184154181362, −1.77113694928293355156939454176, −1.75958434377654984468084238532, −1.43568392168370090733788668972, −1.15232794409164270912100710235, −1.04451124054960848865348154810, −0.31413193449741387225835876737,
0.31413193449741387225835876737, 1.04451124054960848865348154810, 1.15232794409164270912100710235, 1.43568392168370090733788668972, 1.75958434377654984468084238532, 1.77113694928293355156939454176, 2.38877505130440582184154181362, 2.50768481078637230066765402810, 3.09279991862754643048718570422, 3.10424065668116919733448059319, 3.49342218613693284100468809283, 3.58857897256405516883179885675, 4.09759702845009636101798731704, 4.45002159499126000701755496321, 4.48375720134840243426692458265, 4.49908253404808572890183658174, 4.76460831320810299779223629097, 4.97833521983088396506763165975, 5.22013205456007714625647560862, 5.41200375168885157617239214757, 6.04428309238695196799159404662, 6.07829124670115205132284388890, 6.09045037045612896269046654805, 6.20201852156915303392570568172, 6.21979453775535520421196817304
