| L(s) = 1 | − 8·13-s − 8·37-s + 8·41-s − 8·61-s + 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 + ⋯ |
| L(s) = 1 | − 8·13-s − 8·37-s + 8·41-s − 8·61-s + 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3771612077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3771612077\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T^{8} \) |
| good | 3 | \( 1 + T^{16} \) |
| 5 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 41 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{2} )^{8} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.75320897067519182747173004985, −3.43566043374277552351188392069, −3.40630547501026447624257053266, −3.40537319376834261299344118881, −3.18059585951141172184598278916, −3.16936810715799331158382917920, −3.14755882911110498273843616743, −3.14119178614123357838605142706, −2.84265920237594508674792660421, −2.75131713674030097519448378644, −2.41371966081223272242639332675, −2.33651516849284271809292187031, −2.25995878626728419718030403788, −2.17677687779652228402065222307, −2.17660335775675869784094409658, −2.15756330160943100376000468340, −1.98546723564408394121380620457, −1.91119600244485999334217766606, −1.47972338541580235422754902440, −1.43757215197905116700486468789, −1.36678845858280938826912420401, −0.873822057349656624008884387724, −0.75338268380273288994846534574, −0.35169130766545429036568927327, −0.34204834707887138799949925468,
0.34204834707887138799949925468, 0.35169130766545429036568927327, 0.75338268380273288994846534574, 0.873822057349656624008884387724, 1.36678845858280938826912420401, 1.43757215197905116700486468789, 1.47972338541580235422754902440, 1.91119600244485999334217766606, 1.98546723564408394121380620457, 2.15756330160943100376000468340, 2.17660335775675869784094409658, 2.17677687779652228402065222307, 2.25995878626728419718030403788, 2.33651516849284271809292187031, 2.41371966081223272242639332675, 2.75131713674030097519448378644, 2.84265920237594508674792660421, 3.14119178614123357838605142706, 3.14755882911110498273843616743, 3.16936810715799331158382917920, 3.18059585951141172184598278916, 3.40537319376834261299344118881, 3.40630547501026447624257053266, 3.43566043374277552351188392069, 3.75320897067519182747173004985
Plot not available for L-functions of degree greater than 10.
