| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·16-s − 2·17-s − 2·25-s − 2·32-s − 4·34-s + 2·41-s + 2·49-s − 4·50-s − 4·53-s + 3·64-s − 2·68-s + 8·73-s + 81-s + 4·82-s − 2·89-s + 4·98-s − 2·100-s − 8·106-s − 4·109-s − 2·113-s + 127-s + 6·128-s + 131-s + 4·136-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·16-s − 2·17-s − 2·25-s − 2·32-s − 4·34-s + 2·41-s + 2·49-s − 4·50-s − 4·53-s + 3·64-s − 2·68-s + 8·73-s + 81-s + 4·82-s − 2·89-s + 4·98-s − 2·100-s − 8·106-s − 4·109-s − 2·113-s + 127-s + 6·128-s + 131-s + 4·136-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.115337422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115337422\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_1$ | \( ( 1 - T )^{8} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.34931580255574048438690434634, −5.97372076347942185013964865995, −5.68666570645545174058129276191, −5.67939522713324945223301378914, −5.44411905485050849290809274163, −5.24140684206007353563197871498, −5.11440477357892227645320796297, −4.77104393838241323508717218622, −4.67691856139776429122333797382, −4.39586771977096194103928462528, −4.38318603458139826501057660307, −3.93784913287952908489270666964, −3.78262751867943467685731744735, −3.76152245648796587201277002168, −3.70595580251289722373937455033, −3.18120935847587697248391310592, −3.06648288154134137780759846972, −2.59443694387777481537868367253, −2.49133744452317316098971669744, −2.28660932248226151983401421777, −2.19532996977282008249044541779, −1.83513453977715591891449684627, −1.25710205655001143156951588166, −1.01094307114730463950130236101, −0.29449155349676381710278900268,
0.29449155349676381710278900268, 1.01094307114730463950130236101, 1.25710205655001143156951588166, 1.83513453977715591891449684627, 2.19532996977282008249044541779, 2.28660932248226151983401421777, 2.49133744452317316098971669744, 2.59443694387777481537868367253, 3.06648288154134137780759846972, 3.18120935847587697248391310592, 3.70595580251289722373937455033, 3.76152245648796587201277002168, 3.78262751867943467685731744735, 3.93784913287952908489270666964, 4.38318603458139826501057660307, 4.39586771977096194103928462528, 4.67691856139776429122333797382, 4.77104393838241323508717218622, 5.11440477357892227645320796297, 5.24140684206007353563197871498, 5.44411905485050849290809274163, 5.67939522713324945223301378914, 5.68666570645545174058129276191, 5.97372076347942185013964865995, 6.34931580255574048438690434634
