L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s + 2·11-s + 2·13-s + 2·19-s + 9·21-s + 8·23-s + 10·27-s + 2·29-s − 2·31-s + 6·33-s − 4·37-s + 6·39-s + 14·41-s + 12·43-s + 8·47-s + 6·49-s + 14·53-s + 6·57-s + 8·59-s + 2·61-s + 18·63-s + 24·69-s − 6·71-s + 6·73-s + 6·77-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s + 0.603·11-s + 0.554·13-s + 0.458·19-s + 1.96·21-s + 1.66·23-s + 1.92·27-s + 0.371·29-s − 0.359·31-s + 1.04·33-s − 0.657·37-s + 0.960·39-s + 2.18·41-s + 1.82·43-s + 1.16·47-s + 6/7·49-s + 1.92·53-s + 0.794·57-s + 1.04·59-s + 0.256·61-s + 2.26·63-s + 2.88·69-s − 0.712·71-s + 0.702·73-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(18.85172736\) |
\(L(\frac12)\) |
\(\approx\) |
\(18.85172736\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
good | 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 45 T^{2} - 68 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 81 T^{2} + 116 T^{3} + 81 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $D_{6}$ | \( 1 + 4 T + 63 T^{2} + 232 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 712 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 496 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 131 T^{2} - 1012 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 688 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 113 T^{2} + 1052 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 95 T^{2} - 116 T^{3} + 95 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 317 T^{2} + 3096 T^{3} + 317 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 295 T^{2} + 2372 T^{3} + 295 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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
−7.49830807320932891030685331379, −7.20344651368272013663050767288, −7.10681020642436931139586929943, −7.10666109646303555195974130726, −6.27487560566030994595059401690, −6.23355338228539403082611537164, −6.17785235792307531220184315294, −5.53144955574465468595613735524, −5.39863132398323266778724073630, −5.29323806082654797340945239141, −4.66835388354211714474894872135, −4.53803433038942541686713975712, −4.47582151153130186384050806480, −3.87401323785576461394572541035, −3.81754461601042809758104718046, −3.68926053109828767532394821110, −3.16910756899205535055321063300, −2.80635499180977979136856583914, −2.74754745821256026430320461789, −2.20880213122624788908194740924, −2.11990587021749249498655956296, −1.75049036367339217763819388622, −1.13363329568442560291432284888, −0.890511744380821270566633353663, −0.840922612463377632785879951326,
0.840922612463377632785879951326, 0.890511744380821270566633353663, 1.13363329568442560291432284888, 1.75049036367339217763819388622, 2.11990587021749249498655956296, 2.20880213122624788908194740924, 2.74754745821256026430320461789, 2.80635499180977979136856583914, 3.16910756899205535055321063300, 3.68926053109828767532394821110, 3.81754461601042809758104718046, 3.87401323785576461394572541035, 4.47582151153130186384050806480, 4.53803433038942541686713975712, 4.66835388354211714474894872135, 5.29323806082654797340945239141, 5.39863132398323266778724073630, 5.53144955574465468595613735524, 6.17785235792307531220184315294, 6.23355338228539403082611537164, 6.27487560566030994595059401690, 7.10666109646303555195974130726, 7.10681020642436931139586929943, 7.20344651368272013663050767288, 7.49830807320932891030685331379