| L(s) = 1 | − 4-s + 7-s + 13-s − 2·19-s − 25-s − 28-s + 31-s − 2·37-s + 43-s + 49-s − 52-s − 2·61-s + 64-s − 2·67-s + 4·73-s + 2·76-s + 79-s + 91-s + 97-s + 100-s − 2·103-s − 2·109-s − 121-s − 124-s + 127-s + 131-s − 2·133-s + ⋯ |
| L(s) = 1 | − 4-s + 7-s + 13-s − 2·19-s − 25-s − 28-s + 31-s − 2·37-s + 43-s + 49-s − 52-s − 2·61-s + 64-s − 2·67-s + 4·73-s + 2·76-s + 79-s + 91-s + 97-s + 100-s − 2·103-s − 2·109-s − 121-s − 124-s + 127-s + 131-s − 2·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4781627978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4781627978\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.44649465657986661519678677467, −12.20333845879659080516736824876, −11.68700520546869352886476834118, −10.97270749706753325259547122656, −10.60593482334328620591933003771, −10.49979628243254335606059266199, −9.359735561084049327600639408363, −9.339691471161736146011932408401, −8.498228422287741963018967232342, −8.399681262209134790421724523789, −7.88790291501342244301764000697, −7.17565626156071465312996835555, −6.35025518579984016383557468966, −6.12382593642333497846853950934, −5.13768281647080276610406630081, −4.85908750874754404537864165760, −3.93242911116486751096784296401, −3.92544362162399009671072346811, −2.51848137384896730266859745436, −1.62018576048424919760607377832,
1.62018576048424919760607377832, 2.51848137384896730266859745436, 3.92544362162399009671072346811, 3.93242911116486751096784296401, 4.85908750874754404537864165760, 5.13768281647080276610406630081, 6.12382593642333497846853950934, 6.35025518579984016383557468966, 7.17565626156071465312996835555, 7.88790291501342244301764000697, 8.399681262209134790421724523789, 8.498228422287741963018967232342, 9.339691471161736146011932408401, 9.359735561084049327600639408363, 10.49979628243254335606059266199, 10.60593482334328620591933003771, 10.97270749706753325259547122656, 11.68700520546869352886476834118, 12.20333845879659080516736824876, 12.44649465657986661519678677467
