| L(s) = 1 | + 12·9-s − 16·17-s + 40·25-s + 144·41-s + 360·49-s − 80·73-s + 90·81-s − 272·89-s − 528·97-s + 656·113-s − 264·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 192·153-s + 157-s + 163-s + 167-s + 904·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 0.941·17-s + 8/5·25-s + 3.51·41-s + 7.34·49-s − 1.09·73-s + 10/9·81-s − 3.05·89-s − 5.44·97-s + 5.80·113-s − 2.18·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 1.25·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.34·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.04828468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.04828468\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4 p T^{2} - 186 T^{4} - 4 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 12 p T^{2} + 9062 T^{4} + 12 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 452 T^{2} + 102054 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 4 T + 198 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 1092 T^{2} + 534182 T^{4} + 1092 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 516 T^{2} + 998 p^{2} T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1364 T^{2} + 919686 T^{4} - 1364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1012 T^{2} + 112422 T^{4} - 1012 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2372 T^{2} + 5056614 T^{4} - 2372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 36 T + 3302 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 900 T^{2} + 6425702 T^{4} + 900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2116 T^{2} + 7339782 T^{4} - 2116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6292 T^{2} + 19581894 T^{4} - 6292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 5762 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10244 T^{2} + 50952870 T^{4} - 10244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 9156 T^{2} + 41992742 T^{4} + 9156 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 132 T^{2} + 22417862 T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 10 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 6516 T^{2} + 24592550 T^{4} - 6516 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 13636 T^{2} + 114638502 T^{4} + 13636 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 68 T + 15462 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 132 T + 21638 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−4.74325311271771039824936876993, −4.41102206980166782393587239182, −4.40148409664798519766239810780, −4.35415564580773256704174789088, −4.12750220886948030265773666877, −4.10892874441146498788447043225, −4.03952850002777663416098847045, −3.85158081887047015371004729627, −3.75019784939746879223032331572, −3.44417589949482887744243447876, −3.02122709885581896562022428232, −2.98666730153954462294220496686, −2.85337990190841641218506003606, −2.72498663524746244016189514844, −2.59785742039567171972826140660, −2.48245251814859696055176850910, −2.13647827428484131105236094576, −1.85205682552553602393669150735, −1.84763522748759470988221754395, −1.58005507792194573823468831628, −1.22024535837085365784507419111, −0.986987713126649011712094129468, −0.74099886792539981072787112760, −0.55453423134577743180133730555, −0.49227906555588357698196638210,
0.49227906555588357698196638210, 0.55453423134577743180133730555, 0.74099886792539981072787112760, 0.986987713126649011712094129468, 1.22024535837085365784507419111, 1.58005507792194573823468831628, 1.84763522748759470988221754395, 1.85205682552553602393669150735, 2.13647827428484131105236094576, 2.48245251814859696055176850910, 2.59785742039567171972826140660, 2.72498663524746244016189514844, 2.85337990190841641218506003606, 2.98666730153954462294220496686, 3.02122709885581896562022428232, 3.44417589949482887744243447876, 3.75019784939746879223032331572, 3.85158081887047015371004729627, 4.03952850002777663416098847045, 4.10892874441146498788447043225, 4.12750220886948030265773666877, 4.35415564580773256704174789088, 4.40148409664798519766239810780, 4.41102206980166782393587239182, 4.74325311271771039824936876993
Plot not available for L-functions of degree greater than 10.
