| L(s) = 1 | − 4·16-s − 2·81-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 4·16-s − 2·81-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2078877788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2078877788\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 13 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 + T^{2} )^{8} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 97 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
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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
−3.75986085864794475963804474927, −3.64032262501995005123240550149, −3.58273384009855928172905999792, −3.49876983008781627087225229636, −3.24845043026351763888136030405, −3.10586528236397191837339102398, −2.90027751741105085253290193822, −2.84194590857153043208938121814, −2.73402254539154616757548867906, −2.66061930233982610394555566762, −2.61583501372867092449864630700, −2.56290979779887336044222850476, −2.46184614364847612438554450687, −1.99861722418109965796803947369, −1.91255414797703582731105215428, −1.91057408318383392574091427582, −1.86794329364619402023254711624, −1.74152286705688323430625370125, −1.72514242960772954320825151776, −1.46791014383858665754911081906, −1.05172536850922978426754093606, −0.951498143405278233485002290026, −0.76506401512182808837117888086, −0.73440946939596722750165792151, −0.11845158448292587099193066543,
0.11845158448292587099193066543, 0.73440946939596722750165792151, 0.76506401512182808837117888086, 0.951498143405278233485002290026, 1.05172536850922978426754093606, 1.46791014383858665754911081906, 1.72514242960772954320825151776, 1.74152286705688323430625370125, 1.86794329364619402023254711624, 1.91057408318383392574091427582, 1.91255414797703582731105215428, 1.99861722418109965796803947369, 2.46184614364847612438554450687, 2.56290979779887336044222850476, 2.61583501372867092449864630700, 2.66061930233982610394555566762, 2.73402254539154616757548867906, 2.84194590857153043208938121814, 2.90027751741105085253290193822, 3.10586528236397191837339102398, 3.24845043026351763888136030405, 3.49876983008781627087225229636, 3.58273384009855928172905999792, 3.64032262501995005123240550149, 3.75986085864794475963804474927
Plot not available for L-functions of degree greater than 10.
