| L(s) = 1 | + 5·4-s − 20·7-s + 4·13-s + 9·16-s − 2·19-s − 25·25-s − 100·28-s + 40·31-s − 56·37-s + 124·43-s + 202·49-s + 20·52-s − 68·61-s − 35·64-s + 10·67-s + 274·73-s − 10·76-s + 160·79-s − 80·91-s + 34·97-s − 125·100-s + 136·103-s + 232·109-s − 180·112-s + 50·121-s + 200·124-s + 127-s + ⋯ |
| L(s) = 1 | + 5/4·4-s − 2.85·7-s + 4/13·13-s + 9/16·16-s − 0.105·19-s − 25-s − 3.57·28-s + 1.29·31-s − 1.51·37-s + 2.88·43-s + 4.12·49-s + 5/13·52-s − 1.11·61-s − 0.546·64-s + 0.149·67-s + 3.75·73-s − 0.131·76-s + 2.02·79-s − 0.879·91-s + 0.350·97-s − 5/4·100-s + 1.32·103-s + 2.12·109-s − 1.60·112-s + 0.413·121-s + 1.61·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.550432446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550432446\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 470 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 983 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1535 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1535 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1055 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5510 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 10055 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 137 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8978 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 8458 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.33843725620571463601709736235, −11.73364764364532367574937319029, −11.05642151614100974410958455028, −10.68586352062281661959056941971, −10.24313753660192638594704590541, −9.630564608838355934082913897657, −9.467891473993657502232895713788, −8.873760629688658173227519306892, −8.137304335832856400972369939502, −7.39150520434901126398999129714, −7.10191812317339388323527840653, −6.34260581916422716304798451037, −6.28780768087039041146501069963, −5.87984344826735531364812246234, −4.86352657601015975693622326362, −3.73336460715370230349525890841, −3.50917916031197160118166031479, −2.71342331178351756009028559304, −2.18458653345554093888845275006, −0.62691924250033899217861950990,
0.62691924250033899217861950990, 2.18458653345554093888845275006, 2.71342331178351756009028559304, 3.50917916031197160118166031479, 3.73336460715370230349525890841, 4.86352657601015975693622326362, 5.87984344826735531364812246234, 6.28780768087039041146501069963, 6.34260581916422716304798451037, 7.10191812317339388323527840653, 7.39150520434901126398999129714, 8.137304335832856400972369939502, 8.873760629688658173227519306892, 9.467891473993657502232895713788, 9.630564608838355934082913897657, 10.24313753660192638594704590541, 10.68586352062281661959056941971, 11.05642151614100974410958455028, 11.73364764364532367574937319029, 12.33843725620571463601709736235
