| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s + 3·5-s − 9·6-s + 3·7-s − 10·8-s + 6·9-s − 9·10-s − 3·11-s + 18·12-s + 12·13-s − 9·14-s + 9·15-s + 15·16-s + 3·17-s − 18·18-s + 18·20-s + 9·21-s + 9·22-s − 23-s − 30·24-s + 6·25-s − 36·26-s + 10·27-s + 18·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s − 2.84·10-s − 0.904·11-s + 5.19·12-s + 3.32·13-s − 2.40·14-s + 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s + 4.02·20-s + 1.96·21-s + 1.91·22-s − 0.208·23-s − 6.12·24-s + 6/5·25-s − 7.06·26-s + 1.92·27-s + 3.40·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.70132322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.70132322\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 34 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} + 174 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 77 T^{2} + 166 T^{3} + 77 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 83 T^{2} - 170 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 59 T^{2} - 312 T^{3} + 59 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 67 T^{2} - 36 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 21 T + 263 T^{2} - 2054 T^{3} + 263 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 85 T^{2} + 60 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 23 T^{2} + 724 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 125 T^{2} - 488 T^{3} + 125 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 236 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 89 T^{2} - 176 T^{3} + 89 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 143 T^{2} - 456 T^{3} + 143 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} - 392 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 25 T^{2} + 360 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 24 T + 407 T^{2} + 4400 T^{3} + 407 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 37 T^{2} + 1274 T^{3} + 37 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55006127329249243567462246560, −7.07766066150207771149272631236, −6.81477118317156525455305096045, −6.63827598484555361800107812296, −6.30654037836641860395022912589, −6.06099500308826895033894367984, −5.94209498238034936295326747585, −5.49075968901266940469697271108, −5.42976967685011095280415251760, −5.34320096882976147907927875523, −4.49294879100653531332560342708, −4.36342356807936378204085893242, −4.29492020614895836311207899761, −3.60049277446160549741587141088, −3.56996498544484706759516331398, −3.31978007771212236157926726715, −2.86309119054845837324922735592, −2.55457061955656138215221627372, −2.53334425218921434416219016949, −1.87769575839710081705941210228, −1.79917961283997852172047623004, −1.71680298224029938162995745168, −1.07753779739410710419912234535, −0.859596303813084407227525868747, −0.791320477470567192007511747450,
0.791320477470567192007511747450, 0.859596303813084407227525868747, 1.07753779739410710419912234535, 1.71680298224029938162995745168, 1.79917961283997852172047623004, 1.87769575839710081705941210228, 2.53334425218921434416219016949, 2.55457061955656138215221627372, 2.86309119054845837324922735592, 3.31978007771212236157926726715, 3.56996498544484706759516331398, 3.60049277446160549741587141088, 4.29492020614895836311207899761, 4.36342356807936378204085893242, 4.49294879100653531332560342708, 5.34320096882976147907927875523, 5.42976967685011095280415251760, 5.49075968901266940469697271108, 5.94209498238034936295326747585, 6.06099500308826895033894367984, 6.30654037836641860395022912589, 6.63827598484555361800107812296, 6.81477118317156525455305096045, 7.07766066150207771149272631236, 7.55006127329249243567462246560
