| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 2·11-s + 2·13-s − 14-s + 16-s + 3·17-s + 13·19-s − 20-s + 2·22-s + 4·23-s − 2·26-s + 28-s − 29-s − 32-s − 3·34-s − 35-s + 37-s − 13·38-s + 40-s − 7·43-s − 2·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 2.98·19-s − 0.223·20-s + 0.426·22-s + 0.834·23-s − 0.392·26-s + 0.188·28-s − 0.185·29-s − 0.176·32-s − 0.514·34-s − 0.169·35-s + 0.164·37-s − 2.10·38-s + 0.158·40-s − 1.06·43-s − 0.301·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319852658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319852658\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 130 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 187 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−13.4140998260, −12.8968845689, −12.1760515932, −11.9718823473, −11.6695521450, −11.1474410158, −10.7990680301, −10.2962352385, −9.88315323303, −9.39954328077, −9.07471776522, −8.45399762740, −8.06978423635, −7.59788273376, −7.19448321429, −7.00913661600, −5.95595983615, −5.63636569359, −5.18427512086, −4.56655728643, −3.71048384342, −3.17086449260, −2.72983665617, −1.53959417097, −0.880323991209,
0.880323991209, 1.53959417097, 2.72983665617, 3.17086449260, 3.71048384342, 4.56655728643, 5.18427512086, 5.63636569359, 5.95595983615, 7.00913661600, 7.19448321429, 7.59788273376, 8.06978423635, 8.45399762740, 9.07471776522, 9.39954328077, 9.88315323303, 10.2962352385, 10.7990680301, 11.1474410158, 11.6695521450, 11.9718823473, 12.1760515932, 12.8968845689, 13.4140998260
