| L(s) = 1 | − 6·2-s + 27·3-s + 128·4-s − 390·5-s − 162·6-s − 2.08e3·8-s + 2.34e3·10-s + 948·11-s + 3.45e3·12-s − 1.01e4·13-s − 1.05e4·15-s + 1.25e4·16-s − 2.83e4·17-s + 8.62e3·19-s − 4.99e4·20-s − 5.68e3·22-s + 1.52e4·23-s − 5.63e4·24-s + 7.81e4·25-s + 6.11e4·26-s − 1.96e4·27-s + 7.30e4·29-s + 6.31e4·30-s + 2.76e5·31-s − 2.67e5·32-s + 2.55e4·33-s + 1.70e5·34-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 0.577·3-s + 4-s − 1.39·5-s − 0.306·6-s − 1.44·8-s + 0.739·10-s + 0.214·11-s + 0.577·12-s − 1.28·13-s − 0.805·15-s + 0.764·16-s − 1.40·17-s + 0.288·19-s − 1.39·20-s − 0.113·22-s + 0.262·23-s − 0.832·24-s + 25-s + 0.682·26-s − 0.192·27-s + 0.555·29-s + 0.427·30-s + 1.66·31-s − 1.44·32-s + 0.123·33-s + 0.743·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1451194560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1451194560\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 p T - 23 p^{2} T^{2} + 3 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 78 p T + 2959 p^{2} T^{2} + 78 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 948 T - 18588467 T^{2} - 948 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5098 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 28386 T + 395426323 T^{2} + 28386 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 8620 T - 819567339 T^{2} - 8620 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 15288 T - 3171102503 T^{2} - 15288 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 36510 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 276808 T + 49110054753 T^{2} - 276808 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 268526 T - 22825664457 T^{2} + 268526 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 629718 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 685772 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 583296 T - 166388896847 T^{2} + 583296 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 428058 T - 991477488473 T^{2} - 428058 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1306380 T - 782022780419 T^{2} + 1306380 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 300662 T - 3052345197777 T^{2} + 300662 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 507244 T - 5803415129787 T^{2} - 507244 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5560632 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1369082 T - 9173012996373 T^{2} + 1369082 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6913720 T + 28595615252241 T^{2} - 6913720 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4376748 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8528310 T + 28500736560571 T^{2} - 8528310 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8826814 T + p^{7} T^{2} )^{2} \) |
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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
−12.26262708182461601578473697980, −11.49090297472713008032882903574, −11.03045238438490658635132153917, −10.57091286991103013991824576636, −9.795185575927282098895146736116, −9.354061256715192055240419883027, −8.806861036497482316865743257078, −8.242131237703165223035840591183, −7.965537589909133597367514808772, −7.21700292159831811133761862814, −6.64049284666746278450781268092, −6.59185907033658295672714208843, −5.35361601663078105275789186907, −4.74136791384349241964989008100, −3.96559401296087479731310445687, −3.31411372190724584274570724137, −2.60460805341178118409675584720, −2.28500547319412939676208910363, −1.13688477795705688288287603548, −0.11477136610591801454738830990,
0.11477136610591801454738830990, 1.13688477795705688288287603548, 2.28500547319412939676208910363, 2.60460805341178118409675584720, 3.31411372190724584274570724137, 3.96559401296087479731310445687, 4.74136791384349241964989008100, 5.35361601663078105275789186907, 6.59185907033658295672714208843, 6.64049284666746278450781268092, 7.21700292159831811133761862814, 7.965537589909133597367514808772, 8.242131237703165223035840591183, 8.806861036497482316865743257078, 9.354061256715192055240419883027, 9.795185575927282098895146736116, 10.57091286991103013991824576636, 11.03045238438490658635132153917, 11.49090297472713008032882903574, 12.26262708182461601578473697980
