| L(s) = 1 | − 2-s − 4·3-s − 4-s − 5·5-s + 4·6-s − 2·7-s + 3·8-s + 7·9-s + 5·10-s − 6·11-s + 4·12-s − 2·13-s + 2·14-s + 20·15-s − 16-s − 5·17-s − 7·18-s + 4·19-s + 5·20-s + 8·21-s + 6·22-s − 12·24-s + 11·25-s + 2·26-s − 4·27-s + 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1/2·4-s − 2.23·5-s + 1.63·6-s − 0.755·7-s + 1.06·8-s + 7/3·9-s + 1.58·10-s − 1.80·11-s + 1.15·12-s − 0.554·13-s + 0.534·14-s + 5.16·15-s − 1/4·16-s − 1.21·17-s − 1.64·18-s + 0.917·19-s + 1.11·20-s + 1.74·21-s + 1.27·22-s − 2.44·24-s + 11/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 2053 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 151 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T - 40 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−17.1710761647, −16.9265343423, −16.4550597469, −15.8584760423, −15.6562294566, −15.4610640437, −14.4768167990, −13.5201938117, −13.1085781333, −12.4035611246, −12.2308207328, −11.5413176092, −11.1058397552, −10.8232885824, −10.3021934427, −9.58818440756, −8.81683311672, −8.03870857327, −7.49377837262, −7.22378098638, −6.23733950453, −5.43842603697, −4.85385072398, −4.35009577275, −3.24855041430, 0, 0,
3.24855041430, 4.35009577275, 4.85385072398, 5.43842603697, 6.23733950453, 7.22378098638, 7.49377837262, 8.03870857327, 8.81683311672, 9.58818440756, 10.3021934427, 10.8232885824, 11.1058397552, 11.5413176092, 12.2308207328, 12.4035611246, 13.1085781333, 13.5201938117, 14.4768167990, 15.4610640437, 15.6562294566, 15.8584760423, 16.4550597469, 16.9265343423, 17.1710761647
