| L(s) = 1 | + 2-s − 8-s − 11-s − 16-s + 2·17-s − 2·19-s − 22-s − 25-s + 2·34-s − 2·38-s − 41-s + 43-s − 49-s − 50-s − 59-s + 64-s + 67-s − 2·73-s − 82-s + 2·83-s + 86-s + 88-s − 4·89-s + 97-s − 98-s + 2·107-s + 2·113-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s − 11-s − 16-s + 2·17-s − 2·19-s − 22-s − 25-s + 2·34-s − 2·38-s − 41-s + 43-s − 49-s − 50-s − 59-s + 64-s + 67-s − 2·73-s − 82-s + 2·83-s + 86-s + 88-s − 4·89-s + 97-s − 98-s + 2·107-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6819362266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6819362266\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−12.89328439347955743801991004717, −12.43047215731515122245881153187, −12.05278273691497617937970632253, −11.50416066905037812463826442051, −10.89885766179764084292457815165, −10.45068527584334776278651051760, −9.785940083012428641225926651121, −9.666611710270148306101285950825, −8.628332674260032220341229115632, −8.440327092698017982364765255790, −7.78871758008494411283579834540, −7.27330252395249658553950690551, −6.45581861224275953103461660851, −5.85262563460310281494664278878, −5.60946620480892776908866980382, −4.80359899466792150821964388887, −4.34688403368470327698843044606, −3.54587282395174421740539371584, −2.99953893435201362555814620289, −2.02895605873748635540766396231,
2.02895605873748635540766396231, 2.99953893435201362555814620289, 3.54587282395174421740539371584, 4.34688403368470327698843044606, 4.80359899466792150821964388887, 5.60946620480892776908866980382, 5.85262563460310281494664278878, 6.45581861224275953103461660851, 7.27330252395249658553950690551, 7.78871758008494411283579834540, 8.440327092698017982364765255790, 8.628332674260032220341229115632, 9.666611710270148306101285950825, 9.785940083012428641225926651121, 10.45068527584334776278651051760, 10.89885766179764084292457815165, 11.50416066905037812463826442051, 12.05278273691497617937970632253, 12.43047215731515122245881153187, 12.89328439347955743801991004717
