| L(s) = 1 | + 4·2-s + 8·4-s + 24·7-s + 8·8-s + 24·11-s − 48·13-s + 96·14-s − 4·16-s + 24·17-s + 96·22-s + 24·23-s − 192·26-s + 192·28-s − 20·31-s − 32·32-s + 96·34-s − 48·37-s + 192·41-s − 72·43-s + 192·44-s + 96·46-s − 144·47-s + 288·49-s − 384·52-s + 120·53-s + 192·56-s + 44·61-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s + 24/7·7-s + 8-s + 2.18·11-s − 3.69·13-s + 48/7·14-s − 1/4·16-s + 1.41·17-s + 4.36·22-s + 1.04·23-s − 7.38·26-s + 48/7·28-s − 0.645·31-s − 32-s + 2.82·34-s − 1.29·37-s + 4.68·41-s − 1.67·43-s + 4.36·44-s + 2.08·46-s − 3.06·47-s + 5.87·49-s − 7.38·52-s + 2.26·53-s + 24/7·56-s + 0.721·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.91036981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.91036981\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2832 T^{3} + 23087 T^{4} - 2832 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 12 T + 62 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 20640 T^{3} + 301679 T^{4} + 20640 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 6072 T^{3} + 126722 T^{4} - 6072 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 290 T^{2} - 30237 T^{4} - 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} + 8904 T^{3} - 534718 T^{4} + 8904 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2860 T^{2} + 3428358 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 10 T + 3 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 21936 T^{3} - 414046 T^{4} + 21936 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 96 T + 5450 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 174384 T^{3} + 11403839 T^{4} + 174384 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 551376 T^{3} + 26698082 T^{4} + 551376 p^{2} T^{5} + 10368 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 345720 T^{3} + 16595138 T^{4} - 345720 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6062 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 22 T + 4107 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 103632 T^{3} + 37261007 T^{4} - 103632 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 48 T + 10442 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 156720 T^{3} + 17060354 T^{4} + 156720 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 23228 T^{2} + 212771334 T^{4} - 23228 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 90480 T^{3} - 17933566 T^{4} + 90480 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29668 T^{2} + 345034374 T^{4} - 29668 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 2676864 T^{3} + 368210639 T^{4} - 2676864 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−9.300439256601228064200608158191, −8.819228001507169397267765481992, −8.809294021542014801675528609888, −8.654637612646942253111842624159, −7.87443078121951430983014731602, −7.83304998868256779730927861585, −7.40322682557108006869299815136, −7.36958159927804556794473893305, −7.28913821900116935456620475773, −6.56625999275761799856166289984, −6.46189264577079834393651007372, −5.93445090321566060123387600445, −5.46338502344347436070412797616, −5.21891939410534818788712347525, −5.15267374191777388136231061464, −4.75021291357961861678345079788, −4.54411070360325361642009605909, −4.43575994102980985610930978882, −3.82290732683756533395977807398, −3.60481841895861577071402569968, −2.91039669275953596898679997015, −2.49741350280989962356876253428, −2.01763453143321062442793561109, −1.63805437078691561412395395890, −0.996669990122553507709422678391,
0.996669990122553507709422678391, 1.63805437078691561412395395890, 2.01763453143321062442793561109, 2.49741350280989962356876253428, 2.91039669275953596898679997015, 3.60481841895861577071402569968, 3.82290732683756533395977807398, 4.43575994102980985610930978882, 4.54411070360325361642009605909, 4.75021291357961861678345079788, 5.15267374191777388136231061464, 5.21891939410534818788712347525, 5.46338502344347436070412797616, 5.93445090321566060123387600445, 6.46189264577079834393651007372, 6.56625999275761799856166289984, 7.28913821900116935456620475773, 7.36958159927804556794473893305, 7.40322682557108006869299815136, 7.83304998868256779730927861585, 7.87443078121951430983014731602, 8.654637612646942253111842624159, 8.809294021542014801675528609888, 8.819228001507169397267765481992, 9.300439256601228064200608158191
