L(s) = 1 | − 3-s − 5-s − 4·7-s − 2·9-s + 8·11-s − 4·13-s + 15-s + 4·17-s + 4·21-s − 4·23-s − 2·25-s + 2·27-s + 4·29-s − 8·33-s + 4·35-s + 12·37-s + 4·39-s − 12·41-s − 4·43-s + 2·45-s + 4·47-s + 2·49-s − 4·51-s + 4·53-s − 8·55-s − 4·61-s + 8·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s − 2/3·9-s + 2.41·11-s − 1.10·13-s + 0.258·15-s + 0.970·17-s + 0.872·21-s − 0.834·23-s − 2/5·25-s + 0.384·27-s + 0.742·29-s − 1.39·33-s + 0.676·35-s + 1.97·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s + 0.298·45-s + 0.583·47-s + 2/7·49-s − 0.560·51-s + 0.549·53-s − 1.07·55-s − 0.512·61-s + 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4001448407\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4001448407\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.8934316645, −19.6420667034, −19.1551537949, −18.4963041394, −17.6707519500, −17.0080103482, −16.6272610009, −16.5436791565, −15.4732516475, −14.9509208965, −14.1819882616, −13.9008699252, −12.7095155923, −12.3172568607, −11.5820474614, −11.5242621401, −9.96467191573, −9.85184018389, −9.02203563842, −8.09899069409, −6.97238544391, −6.42897107073, −5.70093888867, −4.25303028693, −3.26180587408,
3.26180587408, 4.25303028693, 5.70093888867, 6.42897107073, 6.97238544391, 8.09899069409, 9.02203563842, 9.85184018389, 9.96467191573, 11.5242621401, 11.5820474614, 12.3172568607, 12.7095155923, 13.9008699252, 14.1819882616, 14.9509208965, 15.4732516475, 16.5436791565, 16.6272610009, 17.0080103482, 17.6707519500, 18.4963041394, 19.1551537949, 19.6420667034, 19.8934316645