| L(s) = 1 | − 4-s − 3·5-s + 2·7-s − 2·8-s − 2·9-s + 2·13-s + 16-s + 4·17-s − 3·19-s + 3·20-s + 23-s + 5·25-s − 3·27-s − 2·28-s − 2·29-s + 5·31-s + 4·32-s − 6·35-s + 2·36-s − 10·37-s + 6·40-s + 9·41-s − 2·43-s + 6·45-s + 47-s + 2·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.34·5-s + 0.755·7-s − 0.707·8-s − 2/3·9-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.688·19-s + 0.670·20-s + 0.208·23-s + 25-s − 0.577·27-s − 0.377·28-s − 0.371·29-s + 0.898·31-s + 0.707·32-s − 1.01·35-s + 1/3·36-s − 1.64·37-s + 0.948·40-s + 1.40·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 2/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1038 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1038 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4277040851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4277040851\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 173 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + T - 76 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−19.8142266263, −19.3281057573, −18.8662502165, −18.3710324382, −17.7370925855, −17.0893272217, −16.7481023495, −15.6894699482, −15.4863991706, −14.7857458168, −14.2684359608, −13.7259764250, −12.7072548341, −12.2385549547, −11.5980789743, −11.1414034090, −10.3802549358, −9.32803553476, −8.59975714676, −8.14070673343, −7.45960724565, −6.24792263742, −5.30362282238, −4.24249049646, −3.26670926715,
3.26670926715, 4.24249049646, 5.30362282238, 6.24792263742, 7.45960724565, 8.14070673343, 8.59975714676, 9.32803553476, 10.3802549358, 11.1414034090, 11.5980789743, 12.2385549547, 12.7072548341, 13.7259764250, 14.2684359608, 14.7857458168, 15.4863991706, 15.6894699482, 16.7481023495, 17.0893272217, 17.7370925855, 18.3710324382, 18.8662502165, 19.3281057573, 19.8142266263
