| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 8·17-s − 20·19-s + 2·23-s + 5·25-s − 4·29-s + 12·31-s − 4·35-s + 8·37-s + 4·43-s + 49-s + 24·53-s − 8·55-s + 4·59-s − 18·61-s − 16·67-s + 20·71-s − 8·73-s + 8·77-s + 6·79-s − 4·83-s − 16·85-s + 48·89-s − 40·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 1.94·17-s − 4.58·19-s + 0.417·23-s + 25-s − 0.742·29-s + 2.15·31-s − 0.676·35-s + 1.31·37-s + 0.609·43-s + 1/7·49-s + 3.29·53-s − 1.07·55-s + 0.520·59-s − 2.30·61-s − 1.95·67-s + 2.37·71-s − 0.936·73-s + 0.911·77-s + 0.675·79-s − 0.439·83-s − 1.73·85-s + 5.08·89-s − 4.10·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.168291702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168291702\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 80 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 50 T^{2} + 80 T^{3} + 1819 T^{4} + 80 p T^{5} - 50 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} + 640 T^{3} + 8635 T^{4} + 640 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 107 T^{2} + 90 T^{3} + 11364 T^{4} + 90 p T^{5} - 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.36342228030710423016367168209, −6.08541880532994953123462513614, −5.73285291554331732886684097131, −5.69670983579081659061004437987, −5.34991950241466356671536204392, −5.27343396883740495907600097923, −4.88289543522764685800845048500, −4.61494208318840483764078017493, −4.44680875784617455217026858825, −4.34534740014173472972634064887, −4.31703896368332522902798102218, −4.03345662324417371017886975532, −3.66740409410783654673776884485, −3.43711485177730441972578419761, −3.09993880699324465818559147216, −2.84876759953526749060520700261, −2.53563875895485583929757130106, −2.35730703620098233407785598950, −2.34188251176951576050834979200, −2.09082958122113165159499681266, −1.88651164777162190135914651856, −1.42631884643770872397490246610, −0.937259493279072963151882205814, −0.42828108338245816140965397411, −0.38626747172944161336706018056,
0.38626747172944161336706018056, 0.42828108338245816140965397411, 0.937259493279072963151882205814, 1.42631884643770872397490246610, 1.88651164777162190135914651856, 2.09082958122113165159499681266, 2.34188251176951576050834979200, 2.35730703620098233407785598950, 2.53563875895485583929757130106, 2.84876759953526749060520700261, 3.09993880699324465818559147216, 3.43711485177730441972578419761, 3.66740409410783654673776884485, 4.03345662324417371017886975532, 4.31703896368332522902798102218, 4.34534740014173472972634064887, 4.44680875784617455217026858825, 4.61494208318840483764078017493, 4.88289543522764685800845048500, 5.27343396883740495907600097923, 5.34991950241466356671536204392, 5.69670983579081659061004437987, 5.73285291554331732886684097131, 6.08541880532994953123462513614, 6.36342228030710423016367168209
