| L(s) = 1 | − 8·43-s + 8·103-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 − 2·256-s + ⋯ |
| L(s) = 1 | − 8·43-s + 8·103-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 − 2·256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3326552145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3326552145\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T^{8} \) |
| 13 | \( ( 1 + T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{8} )^{2} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 47 | \( ( 1 + T^{8} )^{2} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{4} \) |
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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
−4.95187547651253930837016627383, −4.75563794454157060964909283617, −4.72050206365518095967900287044, −4.66997268944348490895000365137, −4.63467925979035227287303361122, −4.26182268001566047566648137340, −3.99095436727907082816390778127, −3.80276706457507418806013034548, −3.75146405964570847438051736670, −3.73894123633339661424945036085, −3.59008916874264094497740478474, −3.36298025464857859519113393950, −3.13614529021210741099343827663, −3.10246286632800983312555546459, −3.04133990223048461664853622844, −2.89638044487971571233836669821, −2.41460360310530724210722720962, −2.32369453557618542888148133738, −2.27312621645193045876677843939, −1.97555697114039737323873690728, −1.77830629201225741388919556009, −1.59874041401367885536930546194, −1.51692530949626808455739992371, −1.23527477376490119975142196625, −0.76263624706000351682124792638,
0.76263624706000351682124792638, 1.23527477376490119975142196625, 1.51692530949626808455739992371, 1.59874041401367885536930546194, 1.77830629201225741388919556009, 1.97555697114039737323873690728, 2.27312621645193045876677843939, 2.32369453557618542888148133738, 2.41460360310530724210722720962, 2.89638044487971571233836669821, 3.04133990223048461664853622844, 3.10246286632800983312555546459, 3.13614529021210741099343827663, 3.36298025464857859519113393950, 3.59008916874264094497740478474, 3.73894123633339661424945036085, 3.75146405964570847438051736670, 3.80276706457507418806013034548, 3.99095436727907082816390778127, 4.26182268001566047566648137340, 4.63467925979035227287303361122, 4.66997268944348490895000365137, 4.72050206365518095967900287044, 4.75563794454157060964909283617, 4.95187547651253930837016627383
Plot not available for L-functions of degree greater than 10.
