| L(s) = 1 | + 12·9-s − 680·25-s + 2.50e3·49-s − 3.28e3·73-s − 1.35e3·81-s + 6.16e3·97-s + 5.84e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.71e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 8.16e3·225-s + ⋯ |
| L(s) = 1 | + 4/9·9-s − 5.43·25-s + 7.30·49-s − 5.25·73-s − 1.85·81-s + 6.44·97-s + 4.39·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 7.81·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 2.41·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(16.17869921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.17869921\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 p T^{2} + p^{6} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 34 p T^{2} + p^{6} T^{4} )^{4} \) |
| 7 | \( ( 1 - 626 T^{2} + p^{6} T^{4} )^{4} \) |
| 11 | \( ( 1 - 1462 T^{2} + p^{6} T^{4} )^{4} \) |
| 13 | \( ( 1 - 4294 T^{2} + p^{6} T^{4} )^{4} \) |
| 17 | \( ( 1 - 8546 T^{2} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 + 8858 T^{2} + p^{6} T^{4} )^{4} \) |
| 23 | \( ( 1 + 14926 T^{2} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 25658 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 - 9122 T^{2} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 - 84406 T^{2} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 122162 T^{2} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 + 108554 T^{2} + p^{6} T^{4} )^{4} \) |
| 47 | \( ( 1 + 170014 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 + 74 T^{2} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 - 380758 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 258598 T^{2} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 + 60026 T^{2} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 - 364178 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 + 410 T + p^{3} T^{2} )^{8} \) |
| 79 | \( ( 1 - 978818 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 + 428954 T^{2} + p^{6} T^{4} )^{4} \) |
| 89 | \( ( 1 - 703058 T^{2} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 770 T + p^{3} T^{2} )^{8} \) |
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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
−3.96565838648826127691734493262, −3.93065460574651420009057947090, −3.74736812053309860311665207079, −3.68869052419302128247290635168, −3.56023305061387323061995709750, −3.55127848222355136871795242913, −3.06740028427261753521579229830, −2.97482331177981690835269429390, −2.84311664757298957907149873368, −2.83440222784258863986800771942, −2.54405152122883090771710911091, −2.46214496245086749858585945472, −2.21064387145910842799439186041, −2.00787118761106088620349553122, −1.97839789314912163268942957735, −1.78525301748379057718332843756, −1.77743428813119785004516963842, −1.48804247753461022025947677208, −1.42756009445419981460878955602, −1.04592229680698525170770691345, −0.71085950296386130832905792695, −0.64098665046982865682676463242, −0.53476681375236239435006713265, −0.37558239406808332896279304291, −0.29390635908083038979434211471,
0.29390635908083038979434211471, 0.37558239406808332896279304291, 0.53476681375236239435006713265, 0.64098665046982865682676463242, 0.71085950296386130832905792695, 1.04592229680698525170770691345, 1.42756009445419981460878955602, 1.48804247753461022025947677208, 1.77743428813119785004516963842, 1.78525301748379057718332843756, 1.97839789314912163268942957735, 2.00787118761106088620349553122, 2.21064387145910842799439186041, 2.46214496245086749858585945472, 2.54405152122883090771710911091, 2.83440222784258863986800771942, 2.84311664757298957907149873368, 2.97482331177981690835269429390, 3.06740028427261753521579229830, 3.55127848222355136871795242913, 3.56023305061387323061995709750, 3.68869052419302128247290635168, 3.74736812053309860311665207079, 3.93065460574651420009057947090, 3.96565838648826127691734493262
Plot not available for L-functions of degree greater than 10.
