| L(s) = 1 | − 3·3-s − 3·5-s + 2·7-s + 6·9-s − 4·11-s + 9·15-s + 6·17-s − 4·19-s − 6·21-s − 4·23-s + 6·25-s − 10·27-s − 6·29-s + 6·31-s + 12·33-s − 6·35-s + 4·37-s + 10·41-s − 4·43-s − 18·45-s − 4·47-s − 49-s − 18·51-s − 6·53-s + 12·55-s + 12·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s − 1.20·11-s + 2.32·15-s + 1.45·17-s − 0.917·19-s − 1.30·21-s − 0.834·23-s + 6/5·25-s − 1.92·27-s − 1.11·29-s + 1.07·31-s + 2.08·33-s − 1.01·35-s + 0.657·37-s + 1.56·41-s − 0.609·43-s − 2.68·45-s − 0.583·47-s − 1/7·49-s − 2.52·51-s − 0.824·53-s + 1.61·55-s + 1.58·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7901234891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7901234891\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 9 T^{2} + 24 T^{3} + 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 172 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 136 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 57 T^{2} + 168 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 51 T^{2} - 40 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 912 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 87 T^{2} + 464 T^{3} + 87 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.51318367628323642852153590928, −7.24013251952486591404414878022, −7.08551141347492071997426540032, −6.90188328829830923063996227670, −6.27125856898126271527822100745, −6.19420929508132441580991929513, −6.18287070016915844611106738859, −5.57842873372823564351671767484, −5.53586775544244435373234470831, −5.39892725536632718368534270117, −4.78124191459531774452539218241, −4.71189116424027441962330652355, −4.66906258922285899038024326893, −4.15515507306788425219250652564, −4.02023984959886527610650955107, −3.82336020645274839071274767388, −3.07035863390819746322320735229, −3.04951313327318384325558064777, −2.97472181568849475515390165936, −2.07881884058023212436402243347, −1.81700109318413214431620766155, −1.76094669641838125227420035099, −0.875420136458282469484482080441, −0.74480904316668258954569231580, −0.29432478100593529630591925782,
0.29432478100593529630591925782, 0.74480904316668258954569231580, 0.875420136458282469484482080441, 1.76094669641838125227420035099, 1.81700109318413214431620766155, 2.07881884058023212436402243347, 2.97472181568849475515390165936, 3.04951313327318384325558064777, 3.07035863390819746322320735229, 3.82336020645274839071274767388, 4.02023984959886527610650955107, 4.15515507306788425219250652564, 4.66906258922285899038024326893, 4.71189116424027441962330652355, 4.78124191459531774452539218241, 5.39892725536632718368534270117, 5.53586775544244435373234470831, 5.57842873372823564351671767484, 6.18287070016915844611106738859, 6.19420929508132441580991929513, 6.27125856898126271527822100745, 6.90188328829830923063996227670, 7.08551141347492071997426540032, 7.24013251952486591404414878022, 7.51318367628323642852153590928
