| L(s) = 1 | + 2-s + 4-s − 4·5-s + 2·7-s + 8-s − 4·10-s − 4·13-s + 2·14-s + 16-s + 4·17-s + 8·19-s − 4·20-s + 12·23-s + 3·25-s − 4·26-s + 2·28-s − 8·29-s + 14·31-s + 32-s + 4·34-s − 8·35-s + 8·38-s − 4·40-s − 4·41-s − 4·43-s + 12·46-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s + 0.353·8-s − 1.26·10-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.894·20-s + 2.50·23-s + 3/5·25-s − 0.784·26-s + 0.377·28-s − 1.48·29-s + 2.51·31-s + 0.176·32-s + 0.685·34-s − 1.35·35-s + 1.29·38-s − 0.632·40-s − 0.624·41-s − 0.609·43-s + 1.76·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693823421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693823421\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−14.9436598491, −14.4794814242, −13.9339560031, −13.5135239140, −12.9869991773, −12.3033293989, −11.9899059112, −11.7831081185, −11.2194720778, −11.0227810502, −10.2415148334, −9.51274132652, −9.34482727096, −8.18046494533, −7.82723843094, −7.82476086711, −6.93499396476, −6.71508725989, −5.45678967002, −5.03134570362, −4.77709333112, −3.77979388505, −3.38549178369, −2.63884396406, −1.11568290382,
1.11568290382, 2.63884396406, 3.38549178369, 3.77979388505, 4.77709333112, 5.03134570362, 5.45678967002, 6.71508725989, 6.93499396476, 7.82476086711, 7.82723843094, 8.18046494533, 9.34482727096, 9.51274132652, 10.2415148334, 11.0227810502, 11.2194720778, 11.7831081185, 11.9899059112, 12.3033293989, 12.9869991773, 13.5135239140, 13.9339560031, 14.4794814242, 14.9436598491
