| L(s) = 1 | − 3·2-s − 3·3-s + 2·4-s + 6·5-s + 9·6-s − 2·7-s + 3·8-s − 18·10-s − 3·11-s − 6·12-s + 4·13-s + 6·14-s − 18·15-s − 3·16-s + 22·19-s + 12·20-s + 6·21-s + 9·22-s − 48·23-s − 9·24-s − 25-s − 12·26-s + 27·27-s − 4·28-s + 78·29-s + 54·30-s − 32·31-s + ⋯ |
| L(s) = 1 | − 3/2·2-s − 3-s + 1/2·4-s + 6/5·5-s + 3/2·6-s − 2/7·7-s + 3/8·8-s − 9/5·10-s − 0.272·11-s − 1/2·12-s + 4/13·13-s + 3/7·14-s − 6/5·15-s − 0.187·16-s + 1.15·19-s + 3/5·20-s + 2/7·21-s + 9/22·22-s − 2.08·23-s − 3/8·24-s − 0.0399·25-s − 0.461·26-s + 27-s − 1/7·28-s + 2.68·29-s + 9/5·30-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1927335949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1927335949\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 37 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 48 T + 1297 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 84 T + 4561 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 87 T + 6004 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−21.65185461246845770886572857494, −21.15948344451334266765784998646, −20.05760418643980437206185983639, −19.52798184900777749165917720555, −18.36252858369717876225436281035, −18.15348690502951163348951181700, −17.52416778794563222351021452872, −17.29822143466663229088948166946, −16.17910313472693676683295790448, −15.86169059689982381326924536166, −14.03827374318429537277580525527, −13.83133034304262492305122897953, −12.44252633372168062511752559250, −11.66600372164382799443895964574, −10.25834350509717015183038832534, −10.11159710619766847076183869512, −9.063973502092468953165449586453, −8.114120071487304495902274213765, −6.45974206135036256963934257592, −5.43739916243858562226877482479,
5.43739916243858562226877482479, 6.45974206135036256963934257592, 8.114120071487304495902274213765, 9.063973502092468953165449586453, 10.11159710619766847076183869512, 10.25834350509717015183038832534, 11.66600372164382799443895964574, 12.44252633372168062511752559250, 13.83133034304262492305122897953, 14.03827374318429537277580525527, 15.86169059689982381326924536166, 16.17910313472693676683295790448, 17.29822143466663229088948166946, 17.52416778794563222351021452872, 18.15348690502951163348951181700, 18.36252858369717876225436281035, 19.52798184900777749165917720555, 20.05760418643980437206185983639, 21.15948344451334266765784998646, 21.65185461246845770886572857494
