| L(s) = 1 | − 2-s − 2·4-s − 5-s + 7-s + 3·8-s − 2·9-s + 10-s − 3·11-s − 14-s + 16-s − 3·17-s + 2·18-s − 6·19-s + 2·20-s + 3·22-s + 3·23-s − 2·25-s + 3·27-s − 2·28-s − 17·31-s − 2·32-s + 3·34-s − 35-s + 4·36-s + 6·38-s − 3·40-s − 7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.639·22-s + 0.625·23-s − 2/5·25-s + 0.577·27-s − 0.377·28-s − 3.05·31-s − 0.353·32-s + 0.514·34-s − 0.169·35-s + 2/3·36-s + 0.973·38-s − 0.474·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−16.9022520408, −16.6033486419, −15.8750911565, −15.3589801706, −14.7915586251, −14.5299799519, −13.7423862609, −13.4003262242, −12.8760249759, −12.4180026791, −11.7043487417, −10.9490184874, −10.7186060238, −10.2238135607, −9.18240340140, −8.98322378679, −8.60430763895, −7.85344423968, −7.48661667188, −6.57461008972, −5.63570310853, −5.01331521552, −4.34894466000, −3.52167359876, −2.19061595016, 0,
2.19061595016, 3.52167359876, 4.34894466000, 5.01331521552, 5.63570310853, 6.57461008972, 7.48661667188, 7.85344423968, 8.60430763895, 8.98322378679, 9.18240340140, 10.2238135607, 10.7186060238, 10.9490184874, 11.7043487417, 12.4180026791, 12.8760249759, 13.4003262242, 13.7423862609, 14.5299799519, 14.7915586251, 15.3589801706, 15.8750911565, 16.6033486419, 16.9022520408
