| L(s) = 1 | − 2-s − 2·3-s − 5·5-s + 2·6-s − 8·7-s + 8-s + 4·9-s + 5·10-s − 3·11-s − 2·13-s + 8·14-s + 10·15-s − 16-s − 4·18-s − 4·19-s + 16·21-s + 3·22-s − 23-s − 2·24-s + 10·25-s + 2·26-s − 5·27-s + 3·29-s − 10·30-s + 3·31-s + 6·33-s + 40·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 2.23·5-s + 0.816·6-s − 3.02·7-s + 0.353·8-s + 4/3·9-s + 1.58·10-s − 0.904·11-s − 0.554·13-s + 2.13·14-s + 2.58·15-s − 1/4·16-s − 0.942·18-s − 0.917·19-s + 3.49·21-s + 0.639·22-s − 0.208·23-s − 0.408·24-s + 2·25-s + 0.392·26-s − 0.962·27-s + 0.557·29-s − 1.82·30-s + 0.538·31-s + 1.04·33-s + 6.76·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 1447 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
| good | 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 151 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 18 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T - 101 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 84 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 140 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−16.3288565501, −15.9415201783, −15.7384313813, −15.2772223213, −15.0170943520, −13.6428440713, −13.3348646252, −12.7991444077, −12.3740286206, −12.0310453049, −11.6946206734, −10.8208175241, −10.3769177320, −9.97483314751, −9.68786110069, −8.68211006736, −8.32605989915, −7.47414160434, −7.04432918807, −6.66201085365, −6.03940429087, −5.03949155438, −4.25287432197, −3.63193608664, −2.96017638661, 0, 0,
2.96017638661, 3.63193608664, 4.25287432197, 5.03949155438, 6.03940429087, 6.66201085365, 7.04432918807, 7.47414160434, 8.32605989915, 8.68211006736, 9.68786110069, 9.97483314751, 10.3769177320, 10.8208175241, 11.6946206734, 12.0310453049, 12.3740286206, 12.7991444077, 13.3348646252, 13.6428440713, 15.0170943520, 15.2772223213, 15.7384313813, 15.9415201783, 16.3288565501
