| L(s) = 1 | + 2·2-s + 2·4-s + 5·7-s + 10·14-s − 4·16-s + 4·17-s − 8·23-s − 3·25-s + 10·28-s + 4·31-s − 8·32-s + 8·34-s − 4·41-s − 16·46-s − 12·47-s + 7·49-s − 6·50-s + 8·62-s − 8·64-s + 8·68-s + 12·71-s + 15·73-s + 2·79-s − 8·82-s + 20·89-s − 16·92-s − 24·94-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.88·7-s + 2.67·14-s − 16-s + 0.970·17-s − 1.66·23-s − 3/5·25-s + 1.88·28-s + 0.718·31-s − 1.41·32-s + 1.37·34-s − 0.624·41-s − 2.35·46-s − 1.75·47-s + 49-s − 0.848·50-s + 1.01·62-s − 64-s + 0.970·68-s + 1.42·71-s + 1.75·73-s + 0.225·79-s − 0.883·82-s + 2.11·89-s − 1.66·92-s − 2.47·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.153443819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.153443819\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 99 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−10.17904189359585985710681373976, −9.803868551746532309311564041998, −9.125385151868477579718749863082, −8.316697274412010932138350675137, −7.970925470218798610465494602588, −7.74344607014521610772463115562, −6.71131617038437681658976209441, −6.29759749909325276017130408782, −5.57255124931346416961220074241, −5.08203773178396753695846387460, −4.73219504362817804743756225186, −3.96919083645296628319372199392, −3.46873026827611504364238659080, −2.37697018431639335727003964956, −1.63707256505568160131692061023,
1.63707256505568160131692061023, 2.37697018431639335727003964956, 3.46873026827611504364238659080, 3.96919083645296628319372199392, 4.73219504362817804743756225186, 5.08203773178396753695846387460, 5.57255124931346416961220074241, 6.29759749909325276017130408782, 6.71131617038437681658976209441, 7.74344607014521610772463115562, 7.970925470218798610465494602588, 8.316697274412010932138350675137, 9.125385151868477579718749863082, 9.803868551746532309311564041998, 10.17904189359585985710681373976
