| L(s) = 1 | − 3·2-s − 4·3-s + 6·4-s − 2·5-s + 12·6-s − 3·7-s − 10·8-s + 4·9-s + 6·10-s + 11·11-s − 24·12-s + 9·14-s + 8·15-s + 15·16-s − 2·17-s − 12·18-s + 13·19-s − 12·20-s + 12·21-s − 33·22-s − 7·23-s + 40·24-s − 10·25-s + 9·27-s − 18·28-s − 3·29-s − 24·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 3·4-s − 0.894·5-s + 4.89·6-s − 1.13·7-s − 3.53·8-s + 4/3·9-s + 1.89·10-s + 3.31·11-s − 6.92·12-s + 2.40·14-s + 2.06·15-s + 15/4·16-s − 0.485·17-s − 2.82·18-s + 2.98·19-s − 2.68·20-s + 2.61·21-s − 7.03·22-s − 1.45·23-s + 8.16·24-s − 2·25-s + 1.73·27-s − 3.40·28-s − 0.557·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2711583821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2711583821\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 23 T^{3} + 4 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T + 14 T^{2} + 19 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - p T + 71 T^{2} - 283 T^{3} + 71 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T - 6 T^{2} - 3 T^{3} - 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 13 T + 111 T^{2} - 565 T^{3} + 111 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 273 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 41 T^{2} + 175 T^{3} + 41 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 2 T + 64 T^{2} - 53 T^{3} + 64 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 90 T^{2} + 7 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 7 T + 25 T^{2} - 231 T^{3} + 25 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 17 T + 209 T^{2} + 1533 T^{3} + 209 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 5 T + 77 T^{2} + 499 T^{3} + 77 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 110 T^{2} + 91 T^{3} + 110 p T^{4} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 17 T + 271 T^{2} - 2175 T^{3} + 271 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 137 T^{2} - 367 T^{3} + 137 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 21 T + 327 T^{2} - 3017 T^{3} + 327 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 16 T + 184 T^{2} + 1431 T^{3} + 184 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T + 203 T^{2} + 117 T^{3} + 203 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 10 T + 212 T^{2} - 1455 T^{3} + 212 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - T + 149 T^{2} - 347 T^{3} + 149 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 4 T + 158 T^{2} + 473 T^{3} + 158 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 19 T + 311 T^{2} - 3225 T^{3} + 311 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.084016970868927239143484149566, −7.60551011148246263484782856015, −7.35374733161056425724395300956, −7.32114186186231863650743219982, −6.75032211671209814742780890317, −6.63849992090276419279178304157, −6.57143308921083238014837389108, −6.12699738140616982210206339496, −6.06744560151831436886881709484, −5.89676381345442864684509794807, −5.38996600373743025591202846737, −5.19865218274957874825417263739, −5.10289504619940791190326633301, −4.14562977002813843987151646640, −4.09317204128127398537116563886, −3.94745850858378560479600546269, −3.34112019806461980728366281902, −3.25308847086375744687264018743, −3.02860877027491343033506569454, −2.09571246210658478660392618411, −1.91525654954894085705496731210, −1.54644520637227473966373298927, −0.808789380427477110023012557831, −0.71178262046978744835904803511, −0.38560701585570406103840459407,
0.38560701585570406103840459407, 0.71178262046978744835904803511, 0.808789380427477110023012557831, 1.54644520637227473966373298927, 1.91525654954894085705496731210, 2.09571246210658478660392618411, 3.02860877027491343033506569454, 3.25308847086375744687264018743, 3.34112019806461980728366281902, 3.94745850858378560479600546269, 4.09317204128127398537116563886, 4.14562977002813843987151646640, 5.10289504619940791190326633301, 5.19865218274957874825417263739, 5.38996600373743025591202846737, 5.89676381345442864684509794807, 6.06744560151831436886881709484, 6.12699738140616982210206339496, 6.57143308921083238014837389108, 6.63849992090276419279178304157, 6.75032211671209814742780890317, 7.32114186186231863650743219982, 7.35374733161056425724395300956, 7.60551011148246263484782856015, 8.084016970868927239143484149566
