| L(s) = 1 | − 3·3-s + 3·5-s − 4·7-s + 6·9-s − 6·11-s + 2·13-s − 9·15-s + 2·17-s + 3·19-s + 12·21-s − 6·23-s + 6·25-s − 10·27-s + 18·33-s − 12·35-s − 6·37-s − 6·39-s + 12·41-s + 8·43-s + 18·45-s − 6·47-s − 49-s − 6·51-s + 8·53-s − 18·55-s − 9·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 1.51·7-s + 2·9-s − 1.80·11-s + 0.554·13-s − 2.32·15-s + 0.485·17-s + 0.688·19-s + 2.61·21-s − 1.25·23-s + 6/5·25-s − 1.92·27-s + 3.13·33-s − 2.02·35-s − 0.986·37-s − 0.960·39-s + 1.87·41-s + 1.21·43-s + 2.68·45-s − 0.875·47-s − 1/7·49-s − 0.840·51-s + 1.09·53-s − 2.42·55-s − 1.19·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057778159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057778159\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 21 T^{2} + 56 T^{3} + 21 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 31 T^{2} - 40 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} + 4 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 68 T^{3} + 9 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 63 T^{2} - 36 T^{3} + 63 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 208 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 39 T^{2} + 448 T^{3} + 39 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 988 T^{3} + 147 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 69 T^{2} - 220 T^{3} + 69 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 81 T^{2} + 636 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T - 9 T^{2} + 640 T^{3} - 9 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} + 392 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 227 T^{2} - 1700 T^{3} + 227 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 73 T^{2} - 304 T^{3} + 73 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 16 T + 149 T^{2} + 960 T^{3} + 149 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 167 T^{2} + 1132 T^{3} + 167 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2144 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 229 T^{2} + 308 T^{3} + 229 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 243 T^{2} - 36 T^{3} + 243 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 315 T^{2} + 1952 T^{3} + 315 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.30925108502432286422944268858, −6.92077826475855362974112409661, −6.88086063007551704425284640357, −6.60227655585963548251272259010, −6.20883173791903007207221779654, −5.98197262982819733894319800287, −5.98139400138620937695997937572, −5.63434217598914083511142956427, −5.52700652470099430058778481720, −5.37755533615438930855914775757, −4.87783835188731916326023182473, −4.68892666708127554389538099870, −4.65789177339349409437211297211, −3.80304572312700733500898939915, −3.80233888997935225496187360522, −3.79289267317005557005522078376, −2.99961836171584115217742024377, −2.87160566060930521026877425123, −2.70167708746831260877864579500, −2.20153256751236094641287620428, −1.81425456961360794284694198743, −1.69758718302953965923419867408, −0.870164494203555565654132059981, −0.804103216101608112698240117450, −0.28061460026514670374803679791,
0.28061460026514670374803679791, 0.804103216101608112698240117450, 0.870164494203555565654132059981, 1.69758718302953965923419867408, 1.81425456961360794284694198743, 2.20153256751236094641287620428, 2.70167708746831260877864579500, 2.87160566060930521026877425123, 2.99961836171584115217742024377, 3.79289267317005557005522078376, 3.80233888997935225496187360522, 3.80304572312700733500898939915, 4.65789177339349409437211297211, 4.68892666708127554389538099870, 4.87783835188731916326023182473, 5.37755533615438930855914775757, 5.52700652470099430058778481720, 5.63434217598914083511142956427, 5.98139400138620937695997937572, 5.98197262982819733894319800287, 6.20883173791903007207221779654, 6.60227655585963548251272259010, 6.88086063007551704425284640357, 6.92077826475855362974112409661, 7.30925108502432286422944268858
