| L(s) = 1 | − 2-s + 4-s + 2·5-s + 2·7-s − 8-s − 2·10-s − 6·11-s − 4·13-s − 2·14-s + 16-s + 4·17-s + 8·19-s + 2·20-s + 6·22-s − 3·25-s + 4·26-s + 2·28-s + 4·29-s + 14·31-s − 32-s − 4·34-s + 4·35-s − 8·38-s − 2·40-s − 16·41-s − 4·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.353·8-s − 0.632·10-s − 1.80·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s + 0.447·20-s + 1.27·22-s − 3/5·25-s + 0.784·26-s + 0.377·28-s + 0.742·29-s + 2.51·31-s − 0.176·32-s − 0.685·34-s + 0.676·35-s − 1.29·38-s − 0.316·40-s − 2.49·41-s − 0.609·43-s − 0.904·44-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
| \(L(1)\) |
\(\approx\) |
\(1.153147671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153147671\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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, −14.2102009910, −13.9339560031, −13.5135239140, −13.1830201225, −12.3033293989, −11.9118486001, −11.7831081185, −11.0227810502, −10.2415148334, −10.1152647782, −9.87204525884, −9.34482727096, −8.26804890939, −8.18046494533, −7.82476086711, −6.93499396476, −6.73860900675, −5.45678967002, −5.37363015538, −5.03134570362, −3.77979388505, −2.63884396406, −2.44850978333, −1.11568290382,
1.11568290382, 2.44850978333, 2.63884396406, 3.77979388505, 5.03134570362, 5.37363015538, 5.45678967002, 6.73860900675, 6.93499396476, 7.82476086711, 8.18046494533, 8.26804890939, 9.34482727096, 9.87204525884, 10.1152647782, 10.2415148334, 11.0227810502, 11.7831081185, 11.9118486001, 12.3033293989, 13.1830201225, 13.5135239140, 13.9339560031, 14.2102009910, 14.9436598491
