| L(s) = 1 | − 512·4-s − 6.56e3·9-s + 1.96e5·16-s − 3.15e5·19-s + 2.45e6·31-s + 3.35e6·36-s − 6.72e6·49-s − 6.14e5·61-s − 6.71e7·64-s + 1.61e8·76-s + 3.77e7·79-s + 4.30e7·81-s + 5.42e8·109-s + 4.28e8·121-s − 1.25e9·124-s + 127-s + 131-s + 137-s + 139-s − 1.28e9·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.55e9·169-s + 2.07e9·171-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s − 2.42·19-s + 2.65·31-s + 2·36-s − 1.16·49-s − 0.0444·61-s − 4·64-s + 4.84·76-s + 0.969·79-s + 81-s + 3.84·109-s + 2·121-s − 5.30·124-s − 3·144-s − 1.90·169-s + 2.42·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4444792262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4444792262\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6728927 T^{2} + p^{16} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1554802367 T^{2} + p^{16} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 157967 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1225967 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6771424503358 T^{2} + p^{16} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 23369166652127 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 307393 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 203869623664607 T^{2} + p^{16} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1351474503392638 T^{2} + p^{16} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 18887038 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8929117497058753 T^{2} + p^{16} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.84602224368665758105737761724, −12.61984118786648214305499485028, −12.36066082129345523033311586133, −11.50600657062523265195756699583, −10.88055418122252282360903537850, −10.00925678448787898912855983818, −9.993825920474706422911642402490, −8.940340146754462305710987245906, −8.638733650037724747393695154884, −8.297493832466086433694683231151, −7.66262366289272289562502119769, −6.23324725414672506786935254052, −6.19962336731141950110880663982, −5.02792471881898110227390638107, −4.68175001868724566238201606868, −4.01450302712464015884911069848, −3.24369769566829045571736169136, −2.30812251583507723956150837134, −1.03788330250430650622367001582, −0.26006100188008550248040367308,
0.26006100188008550248040367308, 1.03788330250430650622367001582, 2.30812251583507723956150837134, 3.24369769566829045571736169136, 4.01450302712464015884911069848, 4.68175001868724566238201606868, 5.02792471881898110227390638107, 6.19962336731141950110880663982, 6.23324725414672506786935254052, 7.66262366289272289562502119769, 8.297493832466086433694683231151, 8.638733650037724747393695154884, 8.940340146754462305710987245906, 9.993825920474706422911642402490, 10.00925678448787898912855983818, 10.88055418122252282360903537850, 11.50600657062523265195756699583, 12.36066082129345523033311586133, 12.61984118786648214305499485028, 13.84602224368665758105737761724
