| L(s) = 1 | − 4·2-s + 4·3-s + 8·4-s + 12·5-s − 16·6-s − 8·8-s + 8·9-s − 48·10-s + 4·11-s + 32·12-s + 4·13-s + 48·15-s − 4·16-s − 20·17-s − 32·18-s + 96·20-s − 16·22-s − 40·23-s − 32·24-s + 72·25-s − 16·26-s + 16·27-s − 192·30-s + 48·31-s + 32·32-s + 16·33-s + 80·34-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4/3·3-s + 2·4-s + 12/5·5-s − 8/3·6-s − 8-s + 8/9·9-s − 4.79·10-s + 4/11·11-s + 8/3·12-s + 4/13·13-s + 16/5·15-s − 1/4·16-s − 1.17·17-s − 1.77·18-s + 24/5·20-s − 0.727·22-s − 1.73·23-s − 4/3·24-s + 2.87·25-s − 0.615·26-s + 0.592·27-s − 6.39·30-s + 1.54·31-s + 32-s + 0.484·33-s + 2.35·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.571806012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571806012\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 7 T^{4} - 16 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 19 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 432 T^{3} + 19607 T^{4} - 432 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} - 80 T^{3} - 85817 T^{4} - 80 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 768 T^{2} + 389954 T^{4} - 768 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 1160 T^{3} - 248318 T^{4} + 1160 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 466 T^{2} + 371251 T^{4} - 466 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 24 T + 1940 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T + 5408 T^{2} - 256776 T^{3} + 10981922 T^{4} - 256776 p^{2} T^{5} + 5408 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 2244 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 188 T + 17672 T^{2} - 1157140 T^{3} + 57226414 T^{4} - 1157140 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 84 T + 3528 T^{2} - 51408 T^{3} - 7208953 T^{4} - 51408 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 8444 T^{3} + 8425438 T^{4} - 8444 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7592 T^{2} + 38442738 T^{4} - 7592 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 144 T + 12402 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 92 T + 4232 T^{2} + 445924 T^{3} + 46858654 T^{4} + 445924 p^{2} T^{5} + 4232 p^{4} T^{6} + 92 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 80 T + 11626 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 35208 T^{3} - 22947358 T^{4} + 35208 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8274 T^{2} + 26761235 T^{4} - 8274 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 256 T + 32768 T^{2} - 3746048 T^{3} + 368279842 T^{4} - 3746048 p^{2} T^{5} + 32768 p^{4} T^{6} - 256 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 9442 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 84 T + 3528 T^{2} + 652176 T^{3} + 117853367 T^{4} + 652176 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56880661624584001187598920872, −10.16314882069351075891969671713, −9.791319276545612022631074323349, −9.765658073388702093484877721977, −9.338565862475285429855657219490, −9.146416171599762508978250758068, −9.093387612619428857982522522918, −8.750964311756242616718540265360, −8.312658850172625394897475507556, −8.029099609711865273592174268536, −7.59816041311047606340131438450, −7.54106017659600235503497772340, −7.10536066630070290172979585069, −6.26429508470471472753598419857, −6.20684360766310826984170071506, −6.01386744328284204045872221012, −5.91270698032137994079660177135, −4.76538952926795981896952499426, −4.40466015272455489091663321609, −4.30481718963658307341800393961, −3.16211321174362261438072132431, −2.64755027415312417866694016210, −2.32360048701493387430958811366, −1.79920772946714359039968451075, −1.13651438845308709497281876549,
1.13651438845308709497281876549, 1.79920772946714359039968451075, 2.32360048701493387430958811366, 2.64755027415312417866694016210, 3.16211321174362261438072132431, 4.30481718963658307341800393961, 4.40466015272455489091663321609, 4.76538952926795981896952499426, 5.91270698032137994079660177135, 6.01386744328284204045872221012, 6.20684360766310826984170071506, 6.26429508470471472753598419857, 7.10536066630070290172979585069, 7.54106017659600235503497772340, 7.59816041311047606340131438450, 8.029099609711865273592174268536, 8.312658850172625394897475507556, 8.750964311756242616718540265360, 9.093387612619428857982522522918, 9.146416171599762508978250758068, 9.338565862475285429855657219490, 9.765658073388702093484877721977, 9.791319276545612022631074323349, 10.16314882069351075891969671713, 10.56880661624584001187598920872
