| L(s) = 1 | − 3-s − 4-s − 3·5-s + 7-s + 2·8-s − 3·11-s + 12-s − 2·13-s + 3·15-s + 16-s + 17-s + 5·19-s + 3·20-s − 21-s + 23-s − 2·24-s + 5·25-s + 4·27-s − 28-s − 4·32-s + 3·33-s − 3·35-s − 4·37-s + 2·39-s − 6·40-s − 9·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.707·8-s − 0.904·11-s + 0.288·12-s − 0.554·13-s + 0.774·15-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.670·20-s − 0.218·21-s + 0.208·23-s − 0.408·24-s + 25-s + 0.769·27-s − 0.188·28-s − 0.707·32-s + 0.522·33-s − 0.507·35-s − 0.657·37-s + 0.320·39-s − 0.948·40-s − 1.40·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3486135246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3486135246\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 127 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 16 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.9054817718, −19.3255372690, −18.7707813896, −18.2292872568, −17.7135368063, −16.9790062377, −16.4183971150, −15.9571764584, −15.3067109583, −14.6962325998, −13.9597388923, −13.3185622928, −12.5600821651, −11.8923236171, −11.4770698286, −10.5989964585, −10.1481082508, −9.03337998540, −8.05536019859, −7.69205500554, −6.81393831996, −5.20477015259, −4.83911118217, −3.48602202783,
3.48602202783, 4.83911118217, 5.20477015259, 6.81393831996, 7.69205500554, 8.05536019859, 9.03337998540, 10.1481082508, 10.5989964585, 11.4770698286, 11.8923236171, 12.5600821651, 13.3185622928, 13.9597388923, 14.6962325998, 15.3067109583, 15.9571764584, 16.4183971150, 16.9790062377, 17.7135368063, 18.2292872568, 18.7707813896, 19.3255372690, 19.9054817718
