| L(s) = 1 | − 3-s + 3·4-s − 7-s − 3·12-s + 6·16-s − 17-s + 21-s + 3·25-s − 3·28-s − 29-s − 31-s − 43-s − 47-s − 6·48-s + 51-s − 53-s − 59-s + 10·64-s − 67-s − 3·68-s − 73-s − 3·75-s − 79-s − 83-s + 3·84-s + 87-s − 89-s + ⋯ |
| L(s) = 1 | − 3-s + 3·4-s − 7-s − 3·12-s + 6·16-s − 17-s + 21-s + 3·25-s − 3·28-s − 29-s − 31-s − 43-s − 47-s − 6·48-s + 51-s − 53-s − 59-s + 10·64-s − 67-s − 3·68-s − 73-s − 3·75-s − 79-s − 83-s + 3·84-s + 87-s − 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(587^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(587^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9124204732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9124204732\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 587 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.838666874049892141507433747941, −9.600501392325245834141047380450, −9.244649466909267110176285475324, −8.942806835113643773456308315240, −8.318740297168279214020727920592, −8.301298245034455105910448635684, −7.909994533310226168832253463394, −7.23342102873271238935559842102, −7.21459840707491227948153939833, −7.02845338023650794414654330145, −6.66193839098314404526276452690, −6.43726192515815814491008657425, −6.28013978624111827318829409406, −5.67009880496113991190527435303, −5.63509618626247982236451423849, −5.43335414898550983707438073576, −4.62873582553276218763274380867, −4.52749868076847923963772201697, −3.71951187526017866503408265342, −3.12391539744529558873617571392, −3.10052644376310659719230336950, −2.90147100631987025289221967056, −2.15035607701482456625563808486, −1.75421056129080821426899177957, −1.30030492309883367773104639803,
1.30030492309883367773104639803, 1.75421056129080821426899177957, 2.15035607701482456625563808486, 2.90147100631987025289221967056, 3.10052644376310659719230336950, 3.12391539744529558873617571392, 3.71951187526017866503408265342, 4.52749868076847923963772201697, 4.62873582553276218763274380867, 5.43335414898550983707438073576, 5.63509618626247982236451423849, 5.67009880496113991190527435303, 6.28013978624111827318829409406, 6.43726192515815814491008657425, 6.66193839098314404526276452690, 7.02845338023650794414654330145, 7.21459840707491227948153939833, 7.23342102873271238935559842102, 7.909994533310226168832253463394, 8.301298245034455105910448635684, 8.318740297168279214020727920592, 8.942806835113643773456308315240, 9.244649466909267110176285475324, 9.600501392325245834141047380450, 9.838666874049892141507433747941
