| L(s) = 1 | − 4·9-s − 8·17-s + 26·25-s + 4·41-s − 8·49-s + 88·73-s + 10·81-s − 32·89-s + 112·97-s + 84·113-s + 54·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 1.94·17-s + 26/5·25-s + 0.624·41-s − 8/7·49-s + 10.2·73-s + 10/9·81-s − 3.39·89-s + 11.3·97-s + 7.90·113-s + 4.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.58·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(31.28228629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.28228629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 17 | \( ( 1 + T )^{8} \) |
| good | 5 | \( ( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 27 T^{2} + 416 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T^{2} + 264 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 35 T^{2} + 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 41 T^{2} + 1404 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 88 T^{2} + 3486 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 36 T^{2} + 950 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 11 T^{2} + 1872 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 152 T^{2} + 11550 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 72 T^{2} + 5438 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 208 T^{2} + 19710 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 22 T + 234 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 14 T + p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.49883433206534173990013645525, −3.43240745386328841798426135708, −3.29177403388823157296411740125, −3.25283216655049654238260049259, −3.09056784760186995010282760586, −3.07753153563405034143714713971, −2.86142848588378504723207494709, −2.81693907621653321645051906960, −2.58014127433635261974505404119, −2.46371359773653112197591420436, −2.30209271969079575325015830170, −2.28972673909391245697602619061, −2.12873781690962140181878015216, −2.07384946309360936791082612643, −1.96399471624732725125204395400, −1.77013772366427185524971262569, −1.62873536493018476074239795153, −1.44227188342826731851423699486, −1.25283295401248274185005186750, −0.831015361557121323917898635041, −0.73290320524399181261240145902, −0.65165272953014486850634379918, −0.60533029816068475036378735123, −0.60242863589734796587190692136, −0.45475141624519563395985447465,
0.45475141624519563395985447465, 0.60242863589734796587190692136, 0.60533029816068475036378735123, 0.65165272953014486850634379918, 0.73290320524399181261240145902, 0.831015361557121323917898635041, 1.25283295401248274185005186750, 1.44227188342826731851423699486, 1.62873536493018476074239795153, 1.77013772366427185524971262569, 1.96399471624732725125204395400, 2.07384946309360936791082612643, 2.12873781690962140181878015216, 2.28972673909391245697602619061, 2.30209271969079575325015830170, 2.46371359773653112197591420436, 2.58014127433635261974505404119, 2.81693907621653321645051906960, 2.86142848588378504723207494709, 3.07753153563405034143714713971, 3.09056784760186995010282760586, 3.25283216655049654238260049259, 3.29177403388823157296411740125, 3.43240745386328841798426135708, 3.49883433206534173990013645525
Plot not available for L-functions of degree greater than 10.
