| L(s) = 1 | + 2·3-s − 2·4-s + 4·9-s − 6·11-s − 4·12-s − 4·13-s − 5·16-s + 24·17-s + 4·19-s + 12·23-s + 7·25-s + 4·27-s + 6·29-s − 2·31-s − 12·33-s − 8·36-s − 28·37-s − 8·39-s − 10·43-s + 12·44-s + 12·47-s − 10·48-s + 48·51-s + 8·52-s + 6·53-s + 8·57-s − 36·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 4/3·9-s − 1.80·11-s − 1.15·12-s − 1.10·13-s − 5/4·16-s + 5.82·17-s + 0.917·19-s + 2.50·23-s + 7/5·25-s + 0.769·27-s + 1.11·29-s − 0.359·31-s − 2.08·33-s − 4/3·36-s − 4.60·37-s − 1.28·39-s − 1.52·43-s + 1.80·44-s + 1.75·47-s − 1.44·48-s + 6.72·51-s + 1.10·52-s + 0.824·53-s + 1.05·57-s − 4.68·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.301570476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301570476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 52 T^{3} + 211 T^{4} - 52 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} + 144 T^{3} - 2253 T^{4} + 144 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 31 T^{2} + 234 T^{3} - 228 T^{4} + 234 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 107 T^{2} + 92 T^{3} + 8632 T^{4} + 92 p T^{5} - 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 38 T^{2} - 736 T^{3} - 5213 T^{4} - 736 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.72361798621684757116560259733, −7.47288819113341873011786654496, −7.13680809550509251332853021222, −6.97252280065224466760004065808, −6.96452734630647642697581892214, −6.84920898320166601920928986849, −5.87148668643395276411381107943, −5.83249101278171817777398888436, −5.54088018576110162034431313542, −5.44171517799342129329023548050, −5.07188701256020282566703243885, −5.04452056639603304805067437160, −4.65222864762774614247493968882, −4.53784206997571042454242809246, −4.27938418678846632208569759361, −3.50023209342669921399257422927, −3.40376364954407684761722287047, −3.14921894042439979257103308205, −3.11471394827621832236420945539, −2.80355517524657599539568721867, −2.62151928459227870422242844324, −1.61746147688300150953055964453, −1.47853241552848468257789793826, −1.30317478076645005608008485413, −0.42459196464058160584745715095,
0.42459196464058160584745715095, 1.30317478076645005608008485413, 1.47853241552848468257789793826, 1.61746147688300150953055964453, 2.62151928459227870422242844324, 2.80355517524657599539568721867, 3.11471394827621832236420945539, 3.14921894042439979257103308205, 3.40376364954407684761722287047, 3.50023209342669921399257422927, 4.27938418678846632208569759361, 4.53784206997571042454242809246, 4.65222864762774614247493968882, 5.04452056639603304805067437160, 5.07188701256020282566703243885, 5.44171517799342129329023548050, 5.54088018576110162034431313542, 5.83249101278171817777398888436, 5.87148668643395276411381107943, 6.84920898320166601920928986849, 6.96452734630647642697581892214, 6.97252280065224466760004065808, 7.13680809550509251332853021222, 7.47288819113341873011786654496, 7.72361798621684757116560259733
