| L(s) = 1 | − 2-s − 2·4-s − 2·7-s + 3·8-s + 6·11-s − 6·13-s + 2·14-s + 16-s + 6·17-s − 4·19-s − 6·22-s + 2·23-s + 6·26-s + 4·28-s + 6·29-s − 2·32-s − 6·34-s − 2·37-s + 4·38-s − 2·41-s − 12·44-s − 2·46-s − 6·49-s + 12·52-s − 8·53-s − 6·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 0.755·7-s + 1.06·8-s + 1.80·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 1.27·22-s + 0.417·23-s + 1.17·26-s + 0.755·28-s + 1.11·29-s − 0.353·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.312·41-s − 1.80·44-s − 0.294·46-s − 6/7·49-s + 1.66·52-s − 1.09·53-s − 0.801·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 237 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 247 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 270 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.048591954844025798726995291369, −7.86472972545265933627372795526, −7.29617834333896787556582259290, −6.91362131830000807940518964387, −6.79763031126111179179679411247, −6.33136987266983824397005567538, −5.79498757949898641243831181718, −5.62591157399580600938688328249, −4.96987963072713514762319143876, −4.67403740282486604186818433203, −4.23171145368606743184342150614, −4.18109828154470314620125081303, −3.36370787332793442091661160647, −3.16231352197534681830575698751, −2.72032962600367654451519946729, −2.01161788742329545493641570583, −1.22709857826898618104537923531, −1.20304869746708225638746932687, 0, 0,
1.20304869746708225638746932687, 1.22709857826898618104537923531, 2.01161788742329545493641570583, 2.72032962600367654451519946729, 3.16231352197534681830575698751, 3.36370787332793442091661160647, 4.18109828154470314620125081303, 4.23171145368606743184342150614, 4.67403740282486604186818433203, 4.96987963072713514762319143876, 5.62591157399580600938688328249, 5.79498757949898641243831181718, 6.33136987266983824397005567538, 6.79763031126111179179679411247, 6.91362131830000807940518964387, 7.29617834333896787556582259290, 7.86472972545265933627372795526, 8.048591954844025798726995291369
