| L(s) = 1 | + 4-s − 2·7-s + 6·13-s + 4·16-s − 12·19-s − 20·25-s − 2·28-s + 6·31-s − 10·37-s + 14·43-s − 11·49-s + 6·52-s − 30·61-s + 11·64-s + 2·67-s + 24·73-s − 12·76-s − 22·79-s − 12·91-s + 6·97-s − 20·100-s + 2·109-s − 8·112-s + 4·121-s + 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s + 1.66·13-s + 16-s − 2.75·19-s − 4·25-s − 0.377·28-s + 1.07·31-s − 1.64·37-s + 2.13·43-s − 1.57·49-s + 0.832·52-s − 3.84·61-s + 11/8·64-s + 0.244·67-s + 2.80·73-s − 1.37·76-s − 2.47·79-s − 1.25·91-s + 0.609·97-s − 2·100-s + 0.191·109-s − 0.755·112-s + 4/11·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6710206502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6710206502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 26 T^{2} + 387 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 + 38 T^{2} + 603 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 22 T^{2} - 1197 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 34 T^{2} - 1053 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 86 T^{2} + 4587 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 58 T^{2} - 117 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 106 T^{2} + 4347 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 62 T^{2} - 4077 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.87564911597521824807741565801, −7.66150571889870863861871909153, −7.19436110783332259477414120107, −7.16780385674940629381724572148, −6.47864158540082290693316065992, −6.45329649309305982106050566217, −6.32292614949811262924127221532, −6.21144439715957212112110791888, −5.89120832122382174274368580677, −5.75443296938047430036846845553, −5.43430500912185319282891340992, −5.06915720288724638683226214699, −4.61289914684633452889720694876, −4.52964736841595479585401021137, −3.96735536847710157080667329209, −3.85409000403147554641730157456, −3.75634895091802207344630030279, −3.52091386849219783900137135396, −2.92836040090298976684038372553, −2.73529916463730260517209129607, −2.27372631983631384863324810657, −1.92967121601403283118397270635, −1.64295505736839098282765837996, −1.25713131595285212076105899498, −0.24168670356783388117627951231,
0.24168670356783388117627951231, 1.25713131595285212076105899498, 1.64295505736839098282765837996, 1.92967121601403283118397270635, 2.27372631983631384863324810657, 2.73529916463730260517209129607, 2.92836040090298976684038372553, 3.52091386849219783900137135396, 3.75634895091802207344630030279, 3.85409000403147554641730157456, 3.96735536847710157080667329209, 4.52964736841595479585401021137, 4.61289914684633452889720694876, 5.06915720288724638683226214699, 5.43430500912185319282891340992, 5.75443296938047430036846845553, 5.89120832122382174274368580677, 6.21144439715957212112110791888, 6.32292614949811262924127221532, 6.45329649309305982106050566217, 6.47864158540082290693316065992, 7.16780385674940629381724572148, 7.19436110783332259477414120107, 7.66150571889870863861871909153, 7.87564911597521824807741565801
