| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 3·5-s − 9·6-s − 10·8-s + 6·9-s + 9·10-s + 3·11-s + 18·12-s + 6·13-s − 9·15-s + 15·16-s + 3·17-s − 18·18-s + 6·19-s − 18·20-s − 9·22-s − 6·23-s − 30·24-s + 6·25-s − 18·26-s + 10·27-s + 6·29-s + 27·30-s − 21·32-s + 9·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s − 3.53·8-s + 2·9-s + 2.84·10-s + 0.904·11-s + 5.19·12-s + 1.66·13-s − 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s + 1.37·19-s − 4.02·20-s − 1.91·22-s − 1.25·23-s − 6.12·24-s + 6/5·25-s − 3.53·26-s + 1.92·27-s + 1.11·29-s + 4.92·30-s − 3.71·32-s + 1.56·33-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
| \(L(1)\) |
\(\approx\) |
\(4.271580338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.271580338\) |
| \(L(\frac{3}{2})\) |
|
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 | $A_4\times C_2$ | \( 1 + 9 T^{2} - 8 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 15 T^{2} - 20 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 6 T + 33 T^{2} - 92 T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 252 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 324 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 45 T^{2} + 64 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 792 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1040 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 53 | $A_4\times C_2$ | \( 1 - 12 T + 123 T^{2} - 864 T^{3} + 123 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 12 T + 213 T^{2} + 1440 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 6 T + 147 T^{2} + 580 T^{3} + 147 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 18 T + 261 T^{2} - 2276 T^{3} + 261 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 249 T^{2} - 1728 T^{3} + 249 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T - 21 T^{2} + 988 T^{3} - 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 213 T^{2} - 812 T^{3} + 213 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 89 | $C_6$ | \( 1 + 6 T + 87 T^{2} + 180 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 1112 T^{3} + 147 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.57006201282935731841959725948, −6.93601432162072029837065118485, −6.90717405651552352666159131582, −6.85628378362805759587803291065, −6.33019241886840788402538095582, −6.18504667820549312809415756503, −6.08400593686636590066464182313, −5.57421519392470713549208927390, −5.19676208669575872352954717956, −5.15003248045183010093302783813, −4.46638089068816084238047892456, −4.36695565162260328168036682708, −4.02198601388576552129396280571, −3.60134251879763105098418998302, −3.56363611378848999582728635939, −3.53165298824927998794775453651, −2.92183819647627656909521950869, −2.77250695314359896284256720131, −2.63572999448106467616337422623, −1.95999795917963229250121980846, −1.72040069671895099491386429149, −1.56095787717521117669510821279, −1.03864286342807700427914527605, −0.72265503929385848148597520493, −0.60225196753976481438297452253,
0.60225196753976481438297452253, 0.72265503929385848148597520493, 1.03864286342807700427914527605, 1.56095787717521117669510821279, 1.72040069671895099491386429149, 1.95999795917963229250121980846, 2.63572999448106467616337422623, 2.77250695314359896284256720131, 2.92183819647627656909521950869, 3.53165298824927998794775453651, 3.56363611378848999582728635939, 3.60134251879763105098418998302, 4.02198601388576552129396280571, 4.36695565162260328168036682708, 4.46638089068816084238047892456, 5.15003248045183010093302783813, 5.19676208669575872352954717956, 5.57421519392470713549208927390, 6.08400593686636590066464182313, 6.18504667820549312809415756503, 6.33019241886840788402538095582, 6.85628378362805759587803291065, 6.90717405651552352666159131582, 6.93601432162072029837065118485, 7.57006201282935731841959725948
