| L(s) = 1 | − 2-s − 3-s + 4-s − 5·5-s + 6-s − 4·7-s + 8-s + 5·10-s − 2·11-s − 12-s − 2·13-s + 4·14-s + 5·15-s − 3·16-s + 17-s + 6·19-s − 5·20-s + 4·21-s + 2·22-s − 6·23-s − 24-s + 12·25-s + 2·26-s + 27-s − 4·28-s − 5·30-s − 31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 2.23·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1.58·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1.29·15-s − 3/4·16-s + 0.242·17-s + 1.37·19-s − 1.11·20-s + 0.872·21-s + 0.426·22-s − 1.25·23-s − 0.204·24-s + 12/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s − 0.912·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3798 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3798 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 211 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 84 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 60 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T - T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 81 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 129 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26 T + 343 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
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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
−18.3657796780, −17.6403671687, −16.9411918866, −16.4861737985, −15.9471596302, −15.7634709075, −15.6264495799, −14.6515156346, −13.9378348340, −13.2516643548, −12.4028804449, −12.2588133929, −11.5856855557, −11.2013497522, −10.4881368735, −9.88736057736, −9.37938138453, −8.32201424049, −7.85944768110, −7.30649843830, −6.83086423496, −5.82837333769, −4.77334768819, −3.83921588386, −3.03610267946, 0,
3.03610267946, 3.83921588386, 4.77334768819, 5.82837333769, 6.83086423496, 7.30649843830, 7.85944768110, 8.32201424049, 9.37938138453, 9.88736057736, 10.4881368735, 11.2013497522, 11.5856855557, 12.2588133929, 12.4028804449, 13.2516643548, 13.9378348340, 14.6515156346, 15.6264495799, 15.7634709075, 15.9471596302, 16.4861737985, 16.9411918866, 17.6403671687, 18.3657796780
