| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s + 4·5-s − 9·6-s − 7-s − 10·8-s + 6·9-s − 12·10-s + 2·11-s + 18·12-s + 4·13-s + 3·14-s + 12·15-s + 15·16-s − 3·17-s − 18·18-s − 3·19-s + 24·20-s − 3·21-s − 6·22-s + 9·23-s − 30·24-s + 5·25-s − 12·26-s + 10·27-s − 6·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 1.78·5-s − 3.67·6-s − 0.377·7-s − 3.53·8-s + 2·9-s − 3.79·10-s + 0.603·11-s + 5.19·12-s + 1.10·13-s + 0.801·14-s + 3.09·15-s + 15/4·16-s − 0.727·17-s − 4.24·18-s − 0.688·19-s + 5.36·20-s − 0.654·21-s − 1.27·22-s + 1.87·23-s − 6.12·24-s + 25-s − 2.35·26-s + 1.92·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.919941321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919941321\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 8 T^{2} - 9 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 60 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 84 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 20 T^{2} + 151 T^{3} + 20 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 431 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 83 T^{2} + 112 T^{3} + 83 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 23 T^{2} - 16 T^{3} + 23 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 174 T^{2} - 1125 T^{3} + 174 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 125 T^{2} - 508 T^{3} + 125 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 169 T^{2} + 960 T^{3} + 169 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 74 T^{2} - 43 T^{3} + 74 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 307 T^{2} + 2884 T^{3} + 307 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 22 T + 357 T^{2} + 3304 T^{3} + 357 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1100 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 17 T + 310 T^{2} - 2633 T^{3} + 310 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 81 T^{2} + 544 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 5 T + 100 T^{2} + 183 T^{3} + 100 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 23 T + 406 T^{2} + 4179 T^{3} + 406 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{3} \) |
| 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
−7.31099093477463944497032807648, −6.93441249958741950133038112871, −6.87304385681632248666548337548, −6.49316648140623410775569665947, −6.29730636072305323120403033506, −6.17116042899983955229895516536, −5.96711737475779556465354158217, −5.52212625568954328867684554235, −5.34635942423830407206664625857, −5.16208855501388174226957016109, −4.45466054222496036127383963286, −4.32312063683624028878859304591, −4.28140246732743731295936564247, −3.50075244519821392922104088434, −3.44027200289163152371783785580, −3.37580613013195145341120026160, −2.69799252289266513239183739365, −2.68140200396602120716902944445, −2.48142612283272561415117130439, −2.03066703462556867425278396520, −1.69064151606323225980746350638, −1.58273255296681098936169004989, −1.26123698876824587865549024318, −1.04313554714109963876014217662, −0.23656057361230893229021155932,
0.23656057361230893229021155932, 1.04313554714109963876014217662, 1.26123698876824587865549024318, 1.58273255296681098936169004989, 1.69064151606323225980746350638, 2.03066703462556867425278396520, 2.48142612283272561415117130439, 2.68140200396602120716902944445, 2.69799252289266513239183739365, 3.37580613013195145341120026160, 3.44027200289163152371783785580, 3.50075244519821392922104088434, 4.28140246732743731295936564247, 4.32312063683624028878859304591, 4.45466054222496036127383963286, 5.16208855501388174226957016109, 5.34635942423830407206664625857, 5.52212625568954328867684554235, 5.96711737475779556465354158217, 6.17116042899983955229895516536, 6.29730636072305323120403033506, 6.49316648140623410775569665947, 6.87304385681632248666548337548, 6.93441249958741950133038112871, 7.31099093477463944497032807648
