| L(s) = 1 | − 3·3-s + 4·5-s − 10·7-s + 6·9-s + 11-s − 12·15-s − 7·17-s − 11·19-s + 30·21-s − 2·23-s − 2·25-s − 10·27-s − 8·29-s − 8·31-s − 3·33-s − 40·35-s + 14·37-s + 41-s + 3·43-s + 24·45-s + 9·47-s + 48·49-s + 21·51-s − 13·53-s + 4·55-s + 33·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s − 3.77·7-s + 2·9-s + 0.301·11-s − 3.09·15-s − 1.69·17-s − 2.52·19-s + 6.54·21-s − 0.417·23-s − 2/5·25-s − 1.92·27-s − 1.48·29-s − 1.43·31-s − 0.522·33-s − 6.76·35-s + 2.30·37-s + 0.156·41-s + 0.457·43-s + 3.57·45-s + 1.31·47-s + 48/7·49-s + 2.94·51-s − 1.78·53-s + 0.539·55-s + 4.37·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2032149214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2032149214\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 4 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 10 T + 52 T^{2} + 169 T^{3} + 52 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 3 T^{2} + 21 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 11 T + 67 T^{2} + 305 T^{3} + 67 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 2 T + 26 T^{2} + 175 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 8 T + 70 T^{2} + 299 T^{3} + 70 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 14 T + 174 T^{2} - 1127 T^{3} + 174 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 121 T^{2} - 81 T^{3} + 121 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 3 T + 104 T^{2} - 287 T^{3} + 104 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1596 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 5 T + 179 T^{2} + 573 T^{3} + 179 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 134 T^{2} - 391 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 18 T + 320 T^{2} - 2795 T^{3} + 320 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 215 T^{2} - 1253 T^{3} + 215 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 16 T + 304 T^{2} - 2613 T^{3} + 304 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 5 T + 259 T^{2} + 891 T^{3} + 259 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 5 T + 122 T^{2} + 219 T^{3} + 122 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.72592772762464610537725736689, −6.33959569340576488848413481535, −6.32662396791278320843069232898, −6.27961834629204047044283701557, −6.15902476914723246811740490215, −5.89998827877086359800515247108, −5.83720608021905658061622855080, −5.27822556255449462561312707247, −5.13752811212204484557031219713, −5.08462441173491190600183386260, −4.30410732076659536768203227682, −4.23414292691228230980107177706, −4.17548483887378871039336294142, −3.68702181148244587722155528740, −3.68512769217884323556025675854, −3.40630650297255269975576469639, −2.70412082604389507879881731945, −2.59554233738091422104566754205, −2.46782766306413937563028598130, −1.96664725469812917344432643134, −1.73814507499569493642276852914, −1.73043939602043243517770664676, −0.61628621000940006861106861287, −0.60855949483586979262469890311, −0.14380809828986967109213364828,
0.14380809828986967109213364828, 0.60855949483586979262469890311, 0.61628621000940006861106861287, 1.73043939602043243517770664676, 1.73814507499569493642276852914, 1.96664725469812917344432643134, 2.46782766306413937563028598130, 2.59554233738091422104566754205, 2.70412082604389507879881731945, 3.40630650297255269975576469639, 3.68512769217884323556025675854, 3.68702181148244587722155528740, 4.17548483887378871039336294142, 4.23414292691228230980107177706, 4.30410732076659536768203227682, 5.08462441173491190600183386260, 5.13752811212204484557031219713, 5.27822556255449462561312707247, 5.83720608021905658061622855080, 5.89998827877086359800515247108, 6.15902476914723246811740490215, 6.27961834629204047044283701557, 6.32662396791278320843069232898, 6.33959569340576488848413481535, 6.72592772762464610537725736689
