| L(s) = 1 | − 3·2-s + 8·4-s + 3·5-s + 10·7-s − 45·8-s − 9·10-s − 12·11-s − 8·13-s − 30·14-s + 135·16-s − 252·17-s − 182·19-s + 24·20-s + 36·22-s + 93·23-s + 125·25-s + 24·26-s + 80·28-s − 183·29-s − 170·31-s − 360·32-s + 756·34-s + 30·35-s − 344·37-s + 546·38-s − 135·40-s − 474·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 4-s + 0.268·5-s + 0.539·7-s − 1.98·8-s − 0.284·10-s − 0.328·11-s − 0.170·13-s − 0.572·14-s + 2.10·16-s − 3.59·17-s − 2.19·19-s + 0.268·20-s + 0.348·22-s + 0.843·23-s + 25-s + 0.181·26-s + 0.539·28-s − 1.17·29-s − 0.984·31-s − 1.98·32-s + 3.81·34-s + 0.144·35-s − 1.52·37-s + 2.33·38-s − 0.533·40-s − 1.80·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T - 243 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T - 2133 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 93 T - 3518 T^{2} - 93 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 183 T + 9100 T^{2} + 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 170 T - 891 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 172 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 474 T + 155755 T^{2} + 474 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 p T - 27 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 159 T - 78542 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 603 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 564 T + 112717 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 430 T - 42081 T^{2} - 430 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 439 T - 108042 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 351 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 727 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1232 T + 1024785 T^{2} + 1232 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 p T - 47 p^{2} T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1044 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1151 T + 412128 T^{2} + 1151 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.20411907040420125036625852050, −10.99501472357290762856254701558, −10.41055532035959467869724027483, −10.17352534508705029609261451961, −9.089444228683807519588187122897, −8.894938611204547405078277448156, −8.663066601800610584504962416509, −8.453687445132432117946635335797, −7.21288804955840932155890815252, −6.88456915021573434937278822437, −6.73793342454361195156139951711, −5.80894495782555149168619279890, −5.42962074642695201373345884158, −4.45601791310998974788063589150, −3.98448943949664387844771557655, −2.77045260588171342967608870695, −2.29873832982144754822776257340, −1.71309013009169824042976514907, 0, 0,
1.71309013009169824042976514907, 2.29873832982144754822776257340, 2.77045260588171342967608870695, 3.98448943949664387844771557655, 4.45601791310998974788063589150, 5.42962074642695201373345884158, 5.80894495782555149168619279890, 6.73793342454361195156139951711, 6.88456915021573434937278822437, 7.21288804955840932155890815252, 8.453687445132432117946635335797, 8.663066601800610584504962416509, 8.894938611204547405078277448156, 9.089444228683807519588187122897, 10.17352534508705029609261451961, 10.41055532035959467869724027483, 10.99501472357290762856254701558, 11.20411907040420125036625852050
