| L(s) = 1 | − 2-s + 3-s − 6-s − 2·7-s + 8-s + 9-s + 13-s + 2·14-s − 16-s + 17-s − 18-s − 2·21-s + 23-s + 24-s − 25-s − 26-s + 2·27-s + 29-s − 34-s + 4·37-s + 39-s + 2·42-s − 46-s − 2·47-s − 48-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s − 2·7-s + 8-s + 9-s + 13-s + 2·14-s − 16-s + 17-s − 18-s − 2·21-s + 23-s + 24-s − 25-s − 26-s + 2·27-s + 29-s − 34-s + 4·37-s + 39-s + 2·42-s − 46-s − 2·47-s − 48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8340544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8340544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.067971918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067971918\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.215519605372393620171009883480, −8.811986002893230565674364579946, −8.328063825257339153129734589739, −8.144504080311415666994128250952, −7.86032402923715331889017733720, −7.40840189571551137987850192177, −6.81415880743202642409991221328, −6.68499963886323168403976282472, −6.31225363905851397677899504655, −5.86465715922682318028569953637, −5.36197487237039507766942809976, −4.65619553197791220842898295159, −4.40054473952292998152017747488, −3.87600628666576127716305014746, −3.26816237801106861672720167164, −3.26799955148338571750394874975, −2.58768711863928477175259881496, −2.14781118742825836518892415704, −1.01751438959657666433319557663, −1.01175264422348499062057227176,
1.01175264422348499062057227176, 1.01751438959657666433319557663, 2.14781118742825836518892415704, 2.58768711863928477175259881496, 3.26799955148338571750394874975, 3.26816237801106861672720167164, 3.87600628666576127716305014746, 4.40054473952292998152017747488, 4.65619553197791220842898295159, 5.36197487237039507766942809976, 5.86465715922682318028569953637, 6.31225363905851397677899504655, 6.68499963886323168403976282472, 6.81415880743202642409991221328, 7.40840189571551137987850192177, 7.86032402923715331889017733720, 8.144504080311415666994128250952, 8.328063825257339153129734589739, 8.811986002893230565674364579946, 9.215519605372393620171009883480
