| L(s) = 1 | − 4-s + 46·7-s − 62·13-s − 63·16-s + 214·19-s − 190·25-s − 46·28-s + 442·31-s − 74·37-s + 262·43-s + 901·49-s + 62·52-s + 484·61-s + 127·64-s + 880·67-s − 1.58e3·73-s − 214·76-s − 2.11e3·79-s − 2.85e3·91-s + 2.38e3·97-s + 190·100-s + 2.32e3·103-s − 614·109-s − 2.89e3·112-s − 1.16e3·121-s − 442·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/8·4-s + 2.48·7-s − 1.32·13-s − 0.984·16-s + 2.58·19-s − 1.51·25-s − 0.310·28-s + 2.56·31-s − 0.328·37-s + 0.929·43-s + 2.62·49-s + 0.165·52-s + 1.01·61-s + 0.248·64-s + 1.60·67-s − 2.53·73-s − 0.322·76-s − 3.01·79-s − 3.28·91-s + 2.49·97-s + 0.189·100-s + 2.21·103-s − 0.539·109-s − 2.44·112-s − 0.873·121-s − 0.320·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.240944463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.240944463\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 38 p T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 1162 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 31 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7666 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 107 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22706 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27118 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 221 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 130582 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 131 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 166814 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 102814 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 276182 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 242 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 440 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 624562 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 790 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1057 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 981334 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1288438 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1193 T + p^{3} T^{2} )^{2} \) |
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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
−11.63831073297407112498895028971, −11.59524814979451863157602298680, −11.29485018590500794039564701203, −10.38549854106708506817172253485, −9.884908056913710411572399881333, −9.705996997001208042155340449394, −8.842363900358115583246205041196, −8.463763651947907087416052068262, −7.88319447152100429181823157977, −7.37203819980412269000703100354, −7.36429444653720309591342304276, −6.29472351937307318005980697832, −5.46566693512704797269172415485, −5.17014900855088701923699095373, −4.54680807183752971774759178015, −4.29440706034288240492644398800, −3.07892086018455234516028994319, −2.32760462974796149822347339962, −1.61132571344709172312384039384, −0.77072310191059294722725704252,
0.77072310191059294722725704252, 1.61132571344709172312384039384, 2.32760462974796149822347339962, 3.07892086018455234516028994319, 4.29440706034288240492644398800, 4.54680807183752971774759178015, 5.17014900855088701923699095373, 5.46566693512704797269172415485, 6.29472351937307318005980697832, 7.36429444653720309591342304276, 7.37203819980412269000703100354, 7.88319447152100429181823157977, 8.463763651947907087416052068262, 8.842363900358115583246205041196, 9.705996997001208042155340449394, 9.884908056913710411572399881333, 10.38549854106708506817172253485, 11.29485018590500794039564701203, 11.59524814979451863157602298680, 11.63831073297407112498895028971
