| L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 2·5-s + 2·6-s − 7·7-s + 3·8-s + 2·9-s + 2·10-s − 2·11-s + 4·12-s − 3·13-s + 7·14-s + 4·15-s + 16-s + 17-s − 2·18-s − 10·19-s + 4·20-s + 14·21-s + 2·22-s − 11·23-s − 6·24-s − 7·25-s + 3·26-s − 6·27-s + 14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s − 2.64·7-s + 1.06·8-s + 2/3·9-s + 0.632·10-s − 0.603·11-s + 1.15·12-s − 0.832·13-s + 1.87·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s − 2.29·19-s + 0.894·20-s + 3.05·21-s + 0.426·22-s − 2.29·23-s − 1.22·24-s − 7/5·25-s + 0.588·26-s − 1.15·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16216729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16216729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 4027 | | \( 1+O(T) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 77 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 77 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 15 T + 187 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 145 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 107 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 223 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 255 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.85224436871483197335649174826, −7.50939616460320270353633694237, −7.30974896975348319763846171450, −6.72054655645144530211915249311, −6.44088221649436062100150352183, −5.99677309543797091783548848235, −5.74584922460636842827468475644, −5.45100587679577749404641465168, −4.92909907810301224659728900900, −4.31717581832264436454681995722, −3.84247413746456909930358062807, −3.82404021946333422453026596635, −3.56164333894819981168153762588, −2.67061473860151061007137077991, −2.09389044360463292709620624657, −1.66990175773918597488610417692, 0, 0, 0, 0,
1.66990175773918597488610417692, 2.09389044360463292709620624657, 2.67061473860151061007137077991, 3.56164333894819981168153762588, 3.82404021946333422453026596635, 3.84247413746456909930358062807, 4.31717581832264436454681995722, 4.92909907810301224659728900900, 5.45100587679577749404641465168, 5.74584922460636842827468475644, 5.99677309543797091783548848235, 6.44088221649436062100150352183, 6.72054655645144530211915249311, 7.30974896975348319763846171450, 7.50939616460320270353633694237, 7.85224436871483197335649174826
