| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s + 3·5-s − 9·6-s − 4·7-s − 10·8-s + 6·9-s − 9·10-s − 3·11-s + 18·12-s − 2·13-s + 12·14-s + 9·15-s + 15·16-s − 3·17-s − 18·18-s + 18·20-s − 12·21-s + 9·22-s − 30·24-s + 6·25-s + 6·26-s + 10·27-s − 24·28-s + 2·29-s − 27·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s − 1.51·7-s − 3.53·8-s + 2·9-s − 2.84·10-s − 0.904·11-s + 5.19·12-s − 0.554·13-s + 3.20·14-s + 2.32·15-s + 15/4·16-s − 0.727·17-s − 4.24·18-s + 4.02·20-s − 2.61·21-s + 1.91·22-s − 6.12·24-s + 6/5·25-s + 1.17·26-s + 1.92·27-s − 4.53·28-s + 0.371·29-s − 4.92·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 40 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} + 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 16 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 424 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 91 T^{2} + 604 T^{3} + 91 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 22 T + 263 T^{2} + 2036 T^{3} + 263 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 384 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 155 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 105 T^{2} + 984 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 952 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 173 T^{2} - 832 T^{3} + 173 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 223 T^{2} - 1700 T^{3} + 223 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 141 T^{2} + 1816 T^{3} + 141 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 105 T^{2} + 168 T^{3} + 105 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 247 T^{2} + 1644 T^{3} + 247 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 143 T^{2} - 1460 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.72832874551476421381582102022, −7.23632811943958761170734192764, −7.17279032333976020307162059732, −7.02263922409137200610297869554, −6.70219548981981605078502265807, −6.56916774704361406796130216123, −6.41355499395218221979189702013, −5.96433284287506328228688983756, −5.86127624615706916604045602493, −5.38416720252865626929382430886, −5.15063360449694310933249853341, −4.96654154113639069533511755869, −4.82729858306081329094435465945, −3.98905043802548096450582470089, −3.97422168872793850309625921781, −3.57516570331128827345029416031, −3.26987259707531233577832143560, −3.02956024815038309364799359820, −2.92854973481690527564701287536, −2.38410990198215221996065030861, −2.37961090249296208100346813283, −2.07727975973933848078044485196, −1.60741182858415856380270877750, −1.38929476211244362723425670587, −1.29540655712425178955432848904, 0, 0, 0,
1.29540655712425178955432848904, 1.38929476211244362723425670587, 1.60741182858415856380270877750, 2.07727975973933848078044485196, 2.37961090249296208100346813283, 2.38410990198215221996065030861, 2.92854973481690527564701287536, 3.02956024815038309364799359820, 3.26987259707531233577832143560, 3.57516570331128827345029416031, 3.97422168872793850309625921781, 3.98905043802548096450582470089, 4.82729858306081329094435465945, 4.96654154113639069533511755869, 5.15063360449694310933249853341, 5.38416720252865626929382430886, 5.86127624615706916604045602493, 5.96433284287506328228688983756, 6.41355499395218221979189702013, 6.56916774704361406796130216123, 6.70219548981981605078502265807, 7.02263922409137200610297869554, 7.17279032333976020307162059732, 7.23632811943958761170734192764, 7.72832874551476421381582102022
