| L(s) = 1 | + 8·4-s − 2·16-s + 56·25-s − 128·37-s − 144·43-s − 216·64-s + 208·79-s + 448·100-s + 1.39e3·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s − 1.15e3·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1/8·16-s + 2.23·25-s − 3.45·37-s − 3.34·43-s − 3.37·64-s + 2.63·79-s + 4.47·100-s + 12.7·109-s + 0.396·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6.91·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.378·169-s − 6.69·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.715547255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.715547255\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T^{2} + 25 T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 - 28 T^{2} + 346 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 4082 T^{4} - 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 54562 T^{4} - 32 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 972 T^{2} + 399674 T^{4} - 972 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 372 T^{2} + 27542 T^{4} + 372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 56 p T^{2} + 939922 T^{4} - 56 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1240 T^{2} + 672562 T^{4} - 1240 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 2260 T^{2} + 2591542 T^{4} + 2260 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 32 T + 2598 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 1500 T^{2} + 6099578 T^{4} - 1500 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 36 T + 3978 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 2780 T^{2} + 2542278 T^{4} - 2780 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4368 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 5116 T^{2} + 30707686 T^{4} - 5116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 11240 T^{2} + 59117682 T^{4} + 11240 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 7878 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 18040 T^{2} + 131057362 T^{4} - 18040 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 5368 T^{2} + 62983762 T^{4} + 5368 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 52 T + 12982 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 772 T^{2} + 60681574 T^{4} - 772 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 20700 T^{2} + 208447418 T^{4} - 20700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 34888 T^{2} + 481232722 T^{4} + 34888 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.63260441312978104158362007250, −4.57894318240852526226403451484, −4.51961036249808694881702347183, −4.39217257375137625771290768288, −4.05694296940331791386553749132, −3.83412566742459886146582494377, −3.69013143086517002039324724770, −3.47112391496429903685631388092, −3.21694608723871618902509586253, −3.18920557092942320625101685782, −3.15049273712394011917640356719, −3.14378935699757778661917470631, −3.09283609422992313814105173614, −2.45937715229934090289031712108, −2.33139111143015726300619441927, −2.16684282422724853182672340951, −2.12448067506361637135872232936, −1.92412832647711243531661094943, −1.75990671242383781568439900740, −1.72031063105862583919023356484, −1.46234438255521571059134670389, −0.833939072398442050206583302806, −0.71426826041389769724677715768, −0.70232095954697309994314319481, −0.14670868668597557459712148327,
0.14670868668597557459712148327, 0.70232095954697309994314319481, 0.71426826041389769724677715768, 0.833939072398442050206583302806, 1.46234438255521571059134670389, 1.72031063105862583919023356484, 1.75990671242383781568439900740, 1.92412832647711243531661094943, 2.12448067506361637135872232936, 2.16684282422724853182672340951, 2.33139111143015726300619441927, 2.45937715229934090289031712108, 3.09283609422992313814105173614, 3.14378935699757778661917470631, 3.15049273712394011917640356719, 3.18920557092942320625101685782, 3.21694608723871618902509586253, 3.47112391496429903685631388092, 3.69013143086517002039324724770, 3.83412566742459886146582494377, 4.05694296940331791386553749132, 4.39217257375137625771290768288, 4.51961036249808694881702347183, 4.57894318240852526226403451484, 4.63260441312978104158362007250
Plot not available for L-functions of degree greater than 10.
