| L(s) = 1 | + 2·16-s + 8·31-s + 4·61-s + 4·121-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 + ⋯ |
| L(s) = 1 | + 2·16-s + 8·31-s + 4·61-s + 4·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.913063835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913063835\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T^{4} + T^{8} \) |
| good | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 13 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 - T + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 71 | \( ( 1 + T^{2} )^{8} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 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
−4.32921940695173903882516054259, −4.01210138426191986275298638833, −3.87278746461644785201095187842, −3.83311240615520857392924160459, −3.57347278219466344129823894733, −3.57228677658415322600917372568, −3.55187887649718642116361517769, −3.46911862137714315463332035950, −3.27825115821248863519506582903, −3.05256555670321683265318787858, −2.88198707298615110866155074521, −2.70076653172242960124420367632, −2.65949402407712535131607269334, −2.46631360104468861330299454859, −2.43098139386542800302091404075, −2.39203838478409052843605799611, −2.26155972493654590900433537393, −2.08500688479143862743811439639, −1.58738803157816210270948662783, −1.57703436320421469760729459561, −1.27172041915631068316538179779, −1.09775044191987694214685054986, −1.03032890668133285385043912399, −0.949418929156009399994329756279, −0.792180116850680924500338586414,
0.792180116850680924500338586414, 0.949418929156009399994329756279, 1.03032890668133285385043912399, 1.09775044191987694214685054986, 1.27172041915631068316538179779, 1.57703436320421469760729459561, 1.58738803157816210270948662783, 2.08500688479143862743811439639, 2.26155972493654590900433537393, 2.39203838478409052843605799611, 2.43098139386542800302091404075, 2.46631360104468861330299454859, 2.65949402407712535131607269334, 2.70076653172242960124420367632, 2.88198707298615110866155074521, 3.05256555670321683265318787858, 3.27825115821248863519506582903, 3.46911862137714315463332035950, 3.55187887649718642116361517769, 3.57228677658415322600917372568, 3.57347278219466344129823894733, 3.83311240615520857392924160459, 3.87278746461644785201095187842, 4.01210138426191986275298638833, 4.32921940695173903882516054259
Plot not available for L-functions of degree greater than 10.
