| L(s) = 1 | − 3·2-s + 2·3-s + 4·4-s − 6·6-s − 2·7-s − 3·8-s + 3·9-s + 8·12-s − 4·13-s + 6·14-s + 3·16-s − 9·17-s − 9·18-s − 5·19-s − 4·21-s + 4·23-s − 6·24-s + 12·26-s + 4·27-s − 8·28-s + 12·29-s − 31-s − 6·32-s + 27·34-s + 12·36-s + 8·37-s + 15·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 2·4-s − 2.44·6-s − 0.755·7-s − 1.06·8-s + 9-s + 2.30·12-s − 1.10·13-s + 1.60·14-s + 3/4·16-s − 2.18·17-s − 2.12·18-s − 1.14·19-s − 0.872·21-s + 0.834·23-s − 1.22·24-s + 2.35·26-s + 0.769·27-s − 1.51·28-s + 2.22·29-s − 0.179·31-s − 1.06·32-s + 4.63·34-s + 2·36-s + 1.31·37-s + 2.43·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 85 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 189 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 203 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.60372608362554483712850720457, −7.43639294847814886987386201754, −6.88308819454231372595354721993, −6.88228245927653990424601866527, −6.49337098603901137409534889257, −6.11904092409627536129564158560, −5.66147446236952919986148492630, −5.03299710574885045024086325739, −4.58651281706399231831984716512, −4.44999934184807899258883086890, −3.97935751790111707970984745181, −3.54597263347459943823581575985, −2.90032217138843159561108985932, −2.59895516384515417668132321304, −2.36650101248507999422918268353, −2.08918589068449418306115784433, −1.08516242298008329916884911778, −1.08440815144276054985397107387, 0, 0,
1.08440815144276054985397107387, 1.08516242298008329916884911778, 2.08918589068449418306115784433, 2.36650101248507999422918268353, 2.59895516384515417668132321304, 2.90032217138843159561108985932, 3.54597263347459943823581575985, 3.97935751790111707970984745181, 4.44999934184807899258883086890, 4.58651281706399231831984716512, 5.03299710574885045024086325739, 5.66147446236952919986148492630, 6.11904092409627536129564158560, 6.49337098603901137409534889257, 6.88228245927653990424601866527, 6.88308819454231372595354721993, 7.43639294847814886987386201754, 7.60372608362554483712850720457
