| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s − 4·8-s − 4·9-s + 6·10-s − 4·11-s + 2·12-s − 4·13-s − 3·15-s + 8·16-s − 7·17-s + 8·18-s − 3·19-s − 6·20-s + 8·22-s − 5·23-s − 4·24-s + 2·25-s + 8·26-s − 6·27-s − 29-s + 6·30-s + 13·31-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 1.41·8-s − 4/3·9-s + 1.89·10-s − 1.20·11-s + 0.577·12-s − 1.10·13-s − 0.774·15-s + 2·16-s − 1.69·17-s + 1.88·18-s − 0.688·19-s − 1.34·20-s + 1.70·22-s − 1.04·23-s − 0.816·24-s + 2/5·25-s + 1.56·26-s − 1.15·27-s − 0.185·29-s + 1.09·30-s + 2.33·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100069 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100069 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 100069 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 13 T + 95 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11 T + 90 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 125 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.7783276763, −14.1309613652, −13.6416295546, −13.2296322261, −12.4965927286, −12.1705448178, −11.6449878647, −11.5077316497, −10.9088150137, −10.3375736819, −10.0062890462, −9.41425324254, −8.92030050527, −8.50083197636, −8.15613464915, −7.96847505617, −7.44964330032, −6.65160719462, −6.23984179639, −5.55404878133, −4.74263942000, −4.20052742795, −3.21086112701, −2.74901088905, −2.21837043113, 0, 0,
2.21837043113, 2.74901088905, 3.21086112701, 4.20052742795, 4.74263942000, 5.55404878133, 6.23984179639, 6.65160719462, 7.44964330032, 7.96847505617, 8.15613464915, 8.50083197636, 8.92030050527, 9.41425324254, 10.0062890462, 10.3375736819, 10.9088150137, 11.5077316497, 11.6449878647, 12.1705448178, 12.4965927286, 13.2296322261, 13.6416295546, 14.1309613652, 14.7783276763
