| L(s) = 1 | + 4·2-s + 10·4-s − 2·7-s + 20·8-s − 4·13-s − 8·14-s + 35·16-s − 4·19-s − 6·23-s + 10·25-s − 16·26-s − 20·28-s + 20·31-s + 56·32-s + 8·37-s − 16·38-s − 6·41-s − 4·43-s − 24·46-s + 36·47-s + 7·49-s + 40·50-s − 40·52-s − 40·56-s + 8·61-s + 80·62-s + 84·64-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.755·7-s + 7.07·8-s − 1.10·13-s − 2.13·14-s + 35/4·16-s − 0.917·19-s − 1.25·23-s + 2·25-s − 3.13·26-s − 3.77·28-s + 3.59·31-s + 9.89·32-s + 1.31·37-s − 2.59·38-s − 0.937·41-s − 0.609·43-s − 3.53·46-s + 5.25·47-s + 49-s + 5.65·50-s − 5.54·52-s − 5.34·56-s + 1.02·61-s + 10.1·62-s + 21/2·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.02232020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.02232020\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 56 T^{3} - 233 T^{4} - 56 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 56 T^{3} - 89 T^{4} - 56 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - T^{2} - 54 T^{3} + 12 T^{4} - 54 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 40 T^{2} + 759 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 378 T^{3} - 2796 T^{4} - 378 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 272 T^{3} - 2213 T^{4} - 272 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 18 T + 157 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 24 T + 284 T^{2} + 3024 T^{3} + 30567 T^{4} + 3024 p T^{5} + 284 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 79 T^{2} - 378 T^{3} + 2100 T^{4} - 378 p T^{5} - 79 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 74 T^{2} + 448 T^{3} + 3427 T^{4} + 448 p T^{5} - 74 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.88316519736778680718008100894, −6.71981014442410127222451658635, −6.56176524892409248650293616702, −6.28027290242232149577510658231, −6.01735242760380177635142130855, −5.69912368470237626215955805653, −5.69341668886969722400699768909, −5.55261363985728242311238557284, −5.30651409957930954727202738140, −4.68212732373341461212580019369, −4.60616053109621776014865472143, −4.49402575824741178732324321700, −4.35260855143770727217748518036, −4.27239486488439598139048340080, −3.80201395132030358330795139615, −3.43428084609333900724449221580, −3.37404656139230669622979924582, −2.86599233286990464963645757352, −2.69416100570659515776841478930, −2.48748807354091873390391181193, −2.48119321279969193552510094875, −1.99917534018284401958460990679, −1.55195781323104486028585875814, −0.848425312752751112604036672503, −0.76163926501138266099163050222,
0.76163926501138266099163050222, 0.848425312752751112604036672503, 1.55195781323104486028585875814, 1.99917534018284401958460990679, 2.48119321279969193552510094875, 2.48748807354091873390391181193, 2.69416100570659515776841478930, 2.86599233286990464963645757352, 3.37404656139230669622979924582, 3.43428084609333900724449221580, 3.80201395132030358330795139615, 4.27239486488439598139048340080, 4.35260855143770727217748518036, 4.49402575824741178732324321700, 4.60616053109621776014865472143, 4.68212732373341461212580019369, 5.30651409957930954727202738140, 5.55261363985728242311238557284, 5.69341668886969722400699768909, 5.69912368470237626215955805653, 6.01735242760380177635142130855, 6.28027290242232149577510658231, 6.56176524892409248650293616702, 6.71981014442410127222451658635, 6.88316519736778680718008100894
