| L(s) = 1 | + 2-s + 11-s + 2·17-s + 3·19-s + 22-s − 25-s − 32-s + 2·34-s + 3·38-s + 2·41-s − 2·43-s − 49-s − 50-s − 3·59-s − 64-s − 2·67-s − 2·73-s + 2·82-s − 3·83-s − 2·86-s + 2·89-s + 3·97-s − 98-s + 2·107-s − 3·113-s − 3·118-s + 127-s + ⋯ |
| L(s) = 1 | + 2-s + 11-s + 2·17-s + 3·19-s + 22-s − 25-s − 32-s + 2·34-s + 3·38-s + 2·41-s − 2·43-s − 49-s − 50-s − 3·59-s − 64-s − 2·67-s − 2·73-s + 2·82-s − 3·83-s − 2·86-s + 2·89-s + 3·97-s − 98-s + 2·107-s − 3·113-s − 3·118-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.461045140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461045140\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | | \( 1 \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−7.71180647028480697114936765197, −7.26701973604653971690362006205, −7.22384830096375345010998812755, −7.13352171601269328777069202887, −6.54691432720857762450048863036, −6.30968954399399821354393704193, −6.19026843786343710257042285668, −5.78191450161745878799906852727, −5.73374773485601152840966818515, −5.62774743081545042905803644765, −5.20144151511858762349240667148, −4.93228991655339520492524454961, −4.72520165683107944567022386793, −4.55409991480505825309532335396, −4.41564810886384052064988578427, −3.75162344067135490516729953060, −3.70170857241767895905024759002, −3.40497491778558501238876564787, −3.36488469198821573426710983667, −2.94434209140512884440735261674, −2.69174794412313188263532661467, −2.23198787256045921029426999566, −1.44498777401903465628007817746, −1.36416317590207705578583067242, −1.31857430434776127440643131250,
1.31857430434776127440643131250, 1.36416317590207705578583067242, 1.44498777401903465628007817746, 2.23198787256045921029426999566, 2.69174794412313188263532661467, 2.94434209140512884440735261674, 3.36488469198821573426710983667, 3.40497491778558501238876564787, 3.70170857241767895905024759002, 3.75162344067135490516729953060, 4.41564810886384052064988578427, 4.55409991480505825309532335396, 4.72520165683107944567022386793, 4.93228991655339520492524454961, 5.20144151511858762349240667148, 5.62774743081545042905803644765, 5.73374773485601152840966818515, 5.78191450161745878799906852727, 6.19026843786343710257042285668, 6.30968954399399821354393704193, 6.54691432720857762450048863036, 7.13352171601269328777069202887, 7.22384830096375345010998812755, 7.26701973604653971690362006205, 7.71180647028480697114936765197
