| L(s) = 1 | + 3-s + 2·7-s + 13-s − 2·19-s + 2·21-s − 25-s − 27-s + 2·31-s − 2·37-s + 39-s − 43-s + 49-s − 2·57-s + 61-s − 67-s + 73-s − 75-s − 79-s − 81-s + 2·91-s + 2·93-s − 2·97-s + 2·103-s − 2·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
| L(s) = 1 | + 3-s + 2·7-s + 13-s − 2·19-s + 2·21-s − 25-s − 27-s + 2·31-s − 2·37-s + 39-s − 43-s + 49-s − 2·57-s + 61-s − 67-s + 73-s − 75-s − 79-s − 81-s + 2·91-s + 2·93-s − 2·97-s + 2·103-s − 2·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.567275571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567275571\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43556390435928083917637937386, −10.18564362428135691475151304906, −9.632322595847121007642387963406, −9.071560745922779907350886428329, −8.468149385590209130116238328543, −8.395069155310954657775236370113, −8.314488287632337471742068424673, −7.83193173136075888241536855645, −7.19204268638908730022330304421, −6.75368989737793507852430507160, −6.13454410774604462355546476151, −5.83919761185662911970079025778, −5.03847529746151832884127132371, −4.81220028357752716294396926121, −4.15222513866735705651646429404, −3.83572741687139553627097730059, −3.15760897999164346984201921690, −2.43332388812986961015270571047, −1.87263121972076593513080425191, −1.49051568269942084599327398276,
1.49051568269942084599327398276, 1.87263121972076593513080425191, 2.43332388812986961015270571047, 3.15760897999164346984201921690, 3.83572741687139553627097730059, 4.15222513866735705651646429404, 4.81220028357752716294396926121, 5.03847529746151832884127132371, 5.83919761185662911970079025778, 6.13454410774604462355546476151, 6.75368989737793507852430507160, 7.19204268638908730022330304421, 7.83193173136075888241536855645, 8.314488287632337471742068424673, 8.395069155310954657775236370113, 8.468149385590209130116238328543, 9.071560745922779907350886428329, 9.632322595847121007642387963406, 10.18564362428135691475151304906, 10.43556390435928083917637937386
