| L(s) = 1 | − 4·4-s + 8·9-s + 4·16-s + 20·25-s + 24·29-s − 32·36-s − 72·41-s − 4·49-s + 16·64-s + 4·81-s − 80·100-s + 24·101-s − 96·116-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 32·144-s + 149-s + 151-s + 157-s + 163-s + 288·164-s + 167-s + 8·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s + 8/3·9-s + 16-s + 4·25-s + 4.45·29-s − 5.33·36-s − 11.2·41-s − 4/7·49-s + 2·64-s + 4/9·81-s − 8·100-s + 2.38·101-s − 8.91·116-s − 8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 22.4·164-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07979262131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07979262131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 23 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 9 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 101 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 43 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 139 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.86049569201437676078659633823, −4.67675692465143368471771516070, −4.63503847997264811832697588540, −4.59633351094384728247746388647, −4.42117294655091432941247936253, −4.37788966019256523267579307146, −3.97175262175996624096521352191, −3.81856341845175846131875470047, −3.71291526634921455049241819315, −3.69039147177860266249553216273, −3.44177427987399969342995058303, −3.27463264752123095988364125879, −3.09908376084000593912718895256, −2.91648510559158444814603711166, −2.81982988161587870516636917391, −2.70811422547149365974081392905, −2.44326236908334759821003489889, −1.98218441257278157562948435144, −1.87044681107623145284596938844, −1.61196525242618231958394644837, −1.44694200950501164191740648905, −1.32263048358346555031633610183, −1.05462628239931915036813824325, −0.811455823849379578805745272629, −0.05831176781916417987898484423,
0.05831176781916417987898484423, 0.811455823849379578805745272629, 1.05462628239931915036813824325, 1.32263048358346555031633610183, 1.44694200950501164191740648905, 1.61196525242618231958394644837, 1.87044681107623145284596938844, 1.98218441257278157562948435144, 2.44326236908334759821003489889, 2.70811422547149365974081392905, 2.81982988161587870516636917391, 2.91648510559158444814603711166, 3.09908376084000593912718895256, 3.27463264752123095988364125879, 3.44177427987399969342995058303, 3.69039147177860266249553216273, 3.71291526634921455049241819315, 3.81856341845175846131875470047, 3.97175262175996624096521352191, 4.37788966019256523267579307146, 4.42117294655091432941247936253, 4.59633351094384728247746388647, 4.63503847997264811832697588540, 4.67675692465143368471771516070, 4.86049569201437676078659633823
Plot not available for L-functions of degree greater than 10.
