| L(s) = 1 | − 2·3-s + 2·4-s − 2·5-s + 5·9-s + 6·11-s − 4·12-s + 12·13-s + 4·15-s + 4·16-s + 2·17-s − 12·19-s − 4·20-s − 12·23-s + 25-s − 10·27-s − 12·29-s − 12·31-s − 12·33-s + 10·36-s + 4·37-s − 24·39-s − 8·41-s + 8·43-s + 12·44-s − 10·45-s + 6·47-s − 8·48-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s − 0.894·5-s + 5/3·9-s + 1.80·11-s − 1.15·12-s + 3.32·13-s + 1.03·15-s + 16-s + 0.485·17-s − 2.75·19-s − 0.894·20-s − 2.50·23-s + 1/5·25-s − 1.92·27-s − 2.22·29-s − 2.15·31-s − 2.08·33-s + 5/3·36-s + 0.657·37-s − 3.84·39-s − 1.24·41-s + 1.21·43-s + 1.80·44-s − 1.49·45-s + 0.875·47-s − 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455863251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455863251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 64 T^{2} + 408 T^{3} + 2559 T^{4} + 408 p T^{5} + 64 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 64 T^{2} + 216 T^{3} + 975 T^{4} + 216 p T^{5} + 64 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 44 T^{2} + 56 T^{3} + 1639 T^{4} + 56 p T^{5} - 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6 T - 49 T^{2} + 54 T^{3} + 3324 T^{4} + 54 p T^{5} - 49 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 88 T^{2} - 56 T^{3} + 6391 T^{4} - 56 p T^{5} - 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 114 T^{2} + 9275 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} + 16 T^{3} + 8647 T^{4} + 16 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 74 T^{2} + 147 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 14 T + 61 T^{2} + 322 T^{3} - 3500 T^{4} + 322 p T^{5} + 61 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 18 T + 257 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.935919536074947244906925872393, −8.500521055403966488399657955050, −8.098120991573659252510887467637, −8.049881280374671782188865802418, −7.903793899955661609811754920390, −7.45099874391221177069370324412, −7.05218203393366536299239894391, −7.02984863183736363393922262172, −6.48389545464634259709186269951, −6.42044345362402802223658417884, −6.16613607461706971928875539249, −5.93858818218070980501666393336, −5.74522575030972106312453218398, −5.52474193440927636294836827764, −5.00377842686295782256391078937, −4.29058389914350266383639768552, −4.02831396531318427810135079711, −3.95613539227698334441280996983, −3.88748727199105594600408506683, −3.61054951758320207414868404252, −3.05216437385011477597630498379, −1.86785228511217656602261990802, −1.81565736786526193292900543806, −1.76591004131220072761560198239, −0.72132496680646096448762832860,
0.72132496680646096448762832860, 1.76591004131220072761560198239, 1.81565736786526193292900543806, 1.86785228511217656602261990802, 3.05216437385011477597630498379, 3.61054951758320207414868404252, 3.88748727199105594600408506683, 3.95613539227698334441280996983, 4.02831396531318427810135079711, 4.29058389914350266383639768552, 5.00377842686295782256391078937, 5.52474193440927636294836827764, 5.74522575030972106312453218398, 5.93858818218070980501666393336, 6.16613607461706971928875539249, 6.42044345362402802223658417884, 6.48389545464634259709186269951, 7.02984863183736363393922262172, 7.05218203393366536299239894391, 7.45099874391221177069370324412, 7.903793899955661609811754920390, 8.049881280374671782188865802418, 8.098120991573659252510887467637, 8.500521055403966488399657955050, 8.935919536074947244906925872393
