| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 12·7-s + 4·8-s − 6·9-s + 4·10-s + 4·11-s + 2·13-s − 24·14-s + 8·16-s + 16·17-s − 12·18-s − 12·19-s + 4·20-s + 8·22-s + 12·23-s − 25-s + 4·26-s − 24·28-s − 6·29-s − 8·31-s + 8·32-s + 32·34-s − 24·35-s − 12·36-s − 24·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 4.53·7-s + 1.41·8-s − 2·9-s + 1.26·10-s + 1.20·11-s + 0.554·13-s − 6.41·14-s + 2·16-s + 3.88·17-s − 2.82·18-s − 2.75·19-s + 0.894·20-s + 1.70·22-s + 2.50·23-s − 1/5·25-s + 0.784·26-s − 4.53·28-s − 1.11·29-s − 1.43·31-s + 1.41·32-s + 5.48·34-s − 4.05·35-s − 2·36-s − 3.94·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.896334756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896334756\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + p T^{2} - 14 T^{3} + 16 T^{4} - 14 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 20 T^{3} - 164 T^{4} + 20 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 6 p T^{3} - 1528 T^{4} - 6 p^{2} T^{5} + 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 88 p T^{5} + 13 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 1314 p T^{5} + 181 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 65 T^{2} + 120 T^{3} - 604 T^{4} + 120 p T^{5} + 65 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 88 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 45 T^{2} + 300 T^{3} - 5524 T^{4} + 300 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 702 p T^{5} + 261 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 65 T^{2} + 960 T^{3} - 14284 T^{4} + 960 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 185 T^{2} + 784 T^{3} + 4900 T^{4} + 784 p T^{5} + 185 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.454431403307577527011561784855, −9.431577443043981254048006040540, −9.377429014192716184212136941679, −8.699291742906490160131586632283, −8.674970061269965289316890143279, −8.616955256663735367176161968425, −7.82389321818402387227233563953, −7.42532181144775920698853247253, −7.17584423255542462323549183039, −6.70775926826863527110384410604, −6.70449236918892132066392211244, −6.52813017654069927913816182951, −6.01141520491115696719905698434, −5.65611816523331335781761941739, −5.47860396950598096549537052584, −5.43895307090874604214781066742, −5.36116064237471455353744662941, −3.88642190570248213671668037780, −3.80806570891253224534312301153, −3.80260068693803941041898780837, −3.58274782711945581778145147432, −2.86516634859788755610611831208, −2.81411020733972358810975722088, −2.22104297567126918340592383475, −0.913121273494331531416946209800,
0.913121273494331531416946209800, 2.22104297567126918340592383475, 2.81411020733972358810975722088, 2.86516634859788755610611831208, 3.58274782711945581778145147432, 3.80260068693803941041898780837, 3.80806570891253224534312301153, 3.88642190570248213671668037780, 5.36116064237471455353744662941, 5.43895307090874604214781066742, 5.47860396950598096549537052584, 5.65611816523331335781761941739, 6.01141520491115696719905698434, 6.52813017654069927913816182951, 6.70449236918892132066392211244, 6.70775926826863527110384410604, 7.17584423255542462323549183039, 7.42532181144775920698853247253, 7.82389321818402387227233563953, 8.616955256663735367176161968425, 8.674970061269965289316890143279, 8.699291742906490160131586632283, 9.377429014192716184212136941679, 9.431577443043981254048006040540, 9.454431403307577527011561784855
