| L(s) = 1 | − 6·9-s + 3·11-s − 3·13-s + 6·17-s − 3·19-s + 6·23-s − 27-s − 3·29-s + 15·31-s + 9·37-s − 9·41-s + 3·43-s + 3·47-s − 12·49-s − 3·53-s − 6·59-s − 9·61-s − 3·67-s + 9·71-s − 9·73-s + 12·79-s + 18·81-s + 9·83-s + 6·89-s + 21·97-s − 18·99-s − 12·101-s + ⋯ |
| L(s) = 1 | − 2·9-s + 0.904·11-s − 0.832·13-s + 1.45·17-s − 0.688·19-s + 1.25·23-s − 0.192·27-s − 0.557·29-s + 2.69·31-s + 1.47·37-s − 1.40·41-s + 0.457·43-s + 0.437·47-s − 1.71·49-s − 0.412·53-s − 0.781·59-s − 1.15·61-s − 0.366·67-s + 1.06·71-s − 1.05·73-s + 1.35·79-s + 2·81-s + 0.987·83-s + 0.635·89-s + 2.13·97-s − 1.80·99-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876087230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876087230\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 9 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 27 T^{2} - 49 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 95 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 205 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 187 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 15 T + 159 T^{2} - 1019 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 9 T + 81 T^{2} - 17 p T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 3 T + 120 T^{2} - 239 T^{3} + 120 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 299 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 114 T^{2} + 207 T^{3} + 114 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 412 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 865 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 195 T^{2} + 385 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 192 T^{2} - 1097 T^{3} + 192 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 9 T + 237 T^{2} + 1323 T^{3} + 237 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 12 T + 258 T^{2} - 1825 T^{3} + 258 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 9 T + 147 T^{2} - 1205 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T + 258 T^{2} - 1017 T^{3} + 258 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 381 T^{2} - 4181 T^{3} + 381 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.83618100842938077123073728144, −6.56837453623195357980890306260, −6.50586456247509636999060616743, −6.35815867038480695172792353121, −5.95865589309074845108070136978, −5.80676658364280553685495305069, −5.71566838890555972277452682666, −5.21990688596653368404607764642, −5.00069907792853893194469967999, −4.94721593073318221006844992609, −4.63297581650956690843111760295, −4.31451572698806447668483988503, −4.17556371288345805929063718939, −3.58526717130407054214614880283, −3.54771281741504997072564751630, −3.18023750315256757473232203969, −2.91532897800287268285990493579, −2.73310358318868576908742036066, −2.68385541593646260006988356825, −2.01951643583763247980733042587, −1.85460891445858039149457771225, −1.48028256804947352985528443880, −0.946939581691990867551840971967, −0.71430074116873323829448805729, −0.33242022739631624805323063629,
0.33242022739631624805323063629, 0.71430074116873323829448805729, 0.946939581691990867551840971967, 1.48028256804947352985528443880, 1.85460891445858039149457771225, 2.01951643583763247980733042587, 2.68385541593646260006988356825, 2.73310358318868576908742036066, 2.91532897800287268285990493579, 3.18023750315256757473232203969, 3.54771281741504997072564751630, 3.58526717130407054214614880283, 4.17556371288345805929063718939, 4.31451572698806447668483988503, 4.63297581650956690843111760295, 4.94721593073318221006844992609, 5.00069907792853893194469967999, 5.21990688596653368404607764642, 5.71566838890555972277452682666, 5.80676658364280553685495305069, 5.95865589309074845108070136978, 6.35815867038480695172792353121, 6.50586456247509636999060616743, 6.56837453623195357980890306260, 6.83618100842938077123073728144
