| L(s) = 1 | − 4·2-s − 6·3-s + 8·4-s − 8·5-s + 24·6-s − 12·7-s − 8·8-s + 16·9-s + 32·10-s − 4·11-s − 48·12-s + 48·14-s + 48·15-s − 4·16-s − 6·17-s − 64·18-s − 6·19-s − 64·20-s + 72·21-s + 16·22-s − 6·23-s + 48·24-s + 26·25-s − 24·27-s − 96·28-s − 192·30-s + 32·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 3.46·3-s + 4·4-s − 3.57·5-s + 9.79·6-s − 4.53·7-s − 2.82·8-s + 16/3·9-s + 10.1·10-s − 1.20·11-s − 13.8·12-s + 12.8·14-s + 12.3·15-s − 16-s − 1.45·17-s − 15.0·18-s − 1.37·19-s − 14.3·20-s + 15.7·21-s + 3.41·22-s − 1.25·23-s + 9.79·24-s + 26/5·25-s − 4.61·27-s − 18.1·28-s − 35.0·30-s + 5.65·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1674 T^{3} + 11396 T^{4} + 1674 p T^{5} + 201 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 110 T^{2} - 744 T^{3} + 4059 T^{4} - 744 p T^{5} + 110 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 6291 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} + 32 T^{3} + 7627 T^{4} + 32 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 112 T^{2} + 1404 T^{3} + 18747 T^{4} + 1404 p T^{5} + 112 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 144 T^{3} - 3613 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 2088 T^{3} + 26627 T^{4} - 2088 p T^{5} + 222 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.68541476786181182995949486373, −10.45942435308007440831151171291, −10.32069971646691621635120490554, −9.983686543351787194198763159585, −9.959275773227621639853646702370, −9.367278159471309647633797666211, −8.986403236483997724332395838970, −8.786923276720999196671280305504, −8.740548967189365823097484865780, −7.890101517039449672594169451293, −7.79482891601130147892503056776, −7.78396103446661067938911222170, −7.21601832516443392535352343096, −6.70241883468960821824616526454, −6.67002848047013952318864383646, −6.57648236943004574351513053516, −6.36987589232564461131403420787, −5.69704383225246339218584884348, −5.57173060992550828612275010275, −4.82549975335547338381834631445, −4.31439426193152263666412656159, −4.14899314323525325604034123143, −3.58033679028000759983194361425, −3.28122465418330377340162477398, −2.46386245216985561034633903850, 0, 0, 0, 0,
2.46386245216985561034633903850, 3.28122465418330377340162477398, 3.58033679028000759983194361425, 4.14899314323525325604034123143, 4.31439426193152263666412656159, 4.82549975335547338381834631445, 5.57173060992550828612275010275, 5.69704383225246339218584884348, 6.36987589232564461131403420787, 6.57648236943004574351513053516, 6.67002848047013952318864383646, 6.70241883468960821824616526454, 7.21601832516443392535352343096, 7.78396103446661067938911222170, 7.79482891601130147892503056776, 7.890101517039449672594169451293, 8.740548967189365823097484865780, 8.786923276720999196671280305504, 8.986403236483997724332395838970, 9.367278159471309647633797666211, 9.959275773227621639853646702370, 9.983686543351787194198763159585, 10.32069971646691621635120490554, 10.45942435308007440831151171291, 10.68541476786181182995949486373
