| L(s) = 1 | + 5·4-s − 50·7-s − 122·13-s − 39·16-s + 178·19-s + 86·25-s − 250·28-s − 158·31-s − 614·37-s − 446·43-s + 1.18e3·49-s − 610·52-s − 980·61-s − 515·64-s − 584·67-s − 140·73-s + 890·76-s − 842·79-s + 6.10e3·91-s + 130·97-s + 430·100-s − 1.73e3·103-s − 1.37e3·109-s + 1.95e3·112-s − 2.57e3·121-s − 790·124-s + 127-s + ⋯ |
| L(s) = 1 | + 5/8·4-s − 2.69·7-s − 2.60·13-s − 0.609·16-s + 2.14·19-s + 0.687·25-s − 1.68·28-s − 0.915·31-s − 2.72·37-s − 1.58·43-s + 3.46·49-s − 1.62·52-s − 2.05·61-s − 1.00·64-s − 1.06·67-s − 0.224·73-s + 1.34·76-s − 1.19·79-s + 7.02·91-s + 0.136·97-s + 0.429·100-s − 1.66·103-s − 1.20·109-s + 1.64·112-s − 1.93·121-s − 0.572·124-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 86 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 25 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2578 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 61 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3022 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22234 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 48694 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 79 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 307 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 40738 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 223 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 28310 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 169990 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 147334 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 292 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 624346 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 421 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1091074 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 817234 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 65 T + p^{3} T^{2} )^{2} \) |
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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
−11.85338388366850051573007089893, −10.61628845972150274471915916829, −10.54179830423774248962314427964, −9.783070835116550015859235272913, −9.547699423008785118285335575483, −9.351097189227632609569777045511, −8.661719570178485602210066985158, −7.65958759923020882168759059053, −7.10263825362632012885840928639, −7.03677988897796215947285258531, −6.60407596260981113303185244403, −5.78716581018757084901829977433, −5.24994905690361080229784197408, −4.69428569244584941276658845531, −3.57193727941956160565614974860, −2.97915666306404121185783265444, −2.85573904100308163843856575172, −1.72628717529110994257292179161, 0, 0,
1.72628717529110994257292179161, 2.85573904100308163843856575172, 2.97915666306404121185783265444, 3.57193727941956160565614974860, 4.69428569244584941276658845531, 5.24994905690361080229784197408, 5.78716581018757084901829977433, 6.60407596260981113303185244403, 7.03677988897796215947285258531, 7.10263825362632012885840928639, 7.65958759923020882168759059053, 8.661719570178485602210066985158, 9.351097189227632609569777045511, 9.547699423008785118285335575483, 9.783070835116550015859235272913, 10.54179830423774248962314427964, 10.61628845972150274471915916829, 11.85338388366850051573007089893
