| L(s) = 1 | + 4·7-s + 2·9-s − 12·25-s − 64·43-s + 16·49-s + 8·63-s + 48·67-s + 8·79-s + 2·81-s − 64·109-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s − 48·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2/3·9-s − 2.39·25-s − 9.75·43-s + 16/7·49-s + 1.00·63-s + 5.86·67-s + 0.900·79-s + 2/9·81-s − 6.13·109-s + 2.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s − 3.62·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.568121118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568121118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T^{2} + 330 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 28 T^{2} + 502 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 66 T^{2} + 1794 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2094 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 124 T^{2} + 6934 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 132 T^{2} + 8502 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 78 T^{2} + 7650 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 54 T^{2} + 7338 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 2 T^{2} - 2558 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 244 T^{2} + 29638 T^{4} + 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 140 T^{2} + 10390 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.72528248642961031788294995588, −4.58027477813879873302922164565, −4.12432451271703581053872712623, −4.09867915281595581016623025009, −3.95948460255739213988936241404, −3.95327868946782950008839188570, −3.73208313117000281431988219022, −3.59693169496836312131618964959, −3.51698619393026907329737617703, −3.35235070761860049885774442861, −3.34617176228507953572972277020, −2.98445984295373646221258029883, −2.77346627154386389132364986603, −2.58293177274546726772690592806, −2.56206075953344898461261577944, −2.28477717496569695077260838637, −1.93317304760081195637061469458, −1.93035902403712581330078939215, −1.80688936166653115033064863343, −1.63065455898118870253911559121, −1.48139495975064081710365793604, −1.40928607767552532764558694429, −0.910838874253526255367448860202, −0.49170216858214520740413870832, −0.33403927391182743781231959048,
0.33403927391182743781231959048, 0.49170216858214520740413870832, 0.910838874253526255367448860202, 1.40928607767552532764558694429, 1.48139495975064081710365793604, 1.63065455898118870253911559121, 1.80688936166653115033064863343, 1.93035902403712581330078939215, 1.93317304760081195637061469458, 2.28477717496569695077260838637, 2.56206075953344898461261577944, 2.58293177274546726772690592806, 2.77346627154386389132364986603, 2.98445984295373646221258029883, 3.34617176228507953572972277020, 3.35235070761860049885774442861, 3.51698619393026907329737617703, 3.59693169496836312131618964959, 3.73208313117000281431988219022, 3.95327868946782950008839188570, 3.95948460255739213988936241404, 4.09867915281595581016623025009, 4.12432451271703581053872712623, 4.58027477813879873302922164565, 4.72528248642961031788294995588
Plot not available for L-functions of degree greater than 10.
