| L(s) = 1 | − 2-s − 3·3-s − 4·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 4·10-s − 7·11-s − 3·13-s + 2·14-s + 12·15-s − 16-s − 6·17-s − 4·18-s + 2·19-s + 6·21-s + 7·22-s − 4·23-s − 3·24-s + 7·25-s + 3·26-s − 12·30-s − 8·31-s + 21·33-s + 6·34-s + 8·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.78·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 1.26·10-s − 2.11·11-s − 0.832·13-s + 0.534·14-s + 3.09·15-s − 1/4·16-s − 1.45·17-s − 0.942·18-s + 0.458·19-s + 1.30·21-s + 1.49·22-s − 0.834·23-s − 0.612·24-s + 7/5·25-s + 0.588·26-s − 2.19·30-s − 1.43·31-s + 3.65·33-s + 1.02·34-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20532 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20532 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 39 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 107 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 73 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 17 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−16.2011299678, −15.9107581561, −15.5095519183, −15.1954264991, −14.4577851180, −13.6009461692, −13.0168493492, −12.7342037676, −12.2603217964, −11.6002000556, −11.4327986300, −10.8256507804, −10.4732391047, −10.0254149965, −9.28730874665, −8.62915388159, −7.90974481719, −7.54585564861, −7.15276861963, −6.33009658304, −5.70974205618, −4.90527927356, −4.60413611793, −3.63113450174, −2.55528963638, 0, 0,
2.55528963638, 3.63113450174, 4.60413611793, 4.90527927356, 5.70974205618, 6.33009658304, 7.15276861963, 7.54585564861, 7.90974481719, 8.62915388159, 9.28730874665, 10.0254149965, 10.4732391047, 10.8256507804, 11.4327986300, 11.6002000556, 12.2603217964, 12.7342037676, 13.0168493492, 13.6009461692, 14.4577851180, 15.1954264991, 15.5095519183, 15.9107581561, 16.2011299678
