| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s − 2·16-s + 4·23-s + 20·32-s − 16·46-s − 8·53-s − 32·64-s + 32·92-s + 32·106-s + 8·107-s − 4·113-s + 127-s + 20·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 32·184-s + ⋯ |
| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s − 2·16-s + 4·23-s + 20·32-s − 16·46-s − 8·53-s − 32·64-s + 32·92-s + 32·106-s + 8·107-s − 4·113-s + 127-s + 20·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 32·184-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09527299809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09527299809\) |
| \(L(1)\) |
|
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 | \( ( 1 + T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 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
−4.28736615428135818115492047015, −4.07683681254580985032185196946, −3.92670879058723171195278808026, −3.79161089902089641625625424940, −3.60518447905527237955920397297, −3.48512638490808730413728705295, −3.40389665926964421697743674206, −3.36487174547964856459707515691, −3.04164116748702008826977240626, −2.96577581991373207098789945165, −2.88680937668118379592671039789, −2.63269229092988112320683725222, −2.51670352312665363395238879815, −2.45945762114848560825649365176, −2.35138953418986841020647102354, −1.93768012058986991320882912297, −1.85569189072496565439326249440, −1.74940398504564107309680385852, −1.58762772048819548559872769886, −1.55719241926109969703529944622, −1.29994934193556921827966284901, −1.29118949503361770277262403942, −0.921860638322947005572199690746, −0.861828767446607314889184561646, −0.35181967421435186547135899248,
0.35181967421435186547135899248, 0.861828767446607314889184561646, 0.921860638322947005572199690746, 1.29118949503361770277262403942, 1.29994934193556921827966284901, 1.55719241926109969703529944622, 1.58762772048819548559872769886, 1.74940398504564107309680385852, 1.85569189072496565439326249440, 1.93768012058986991320882912297, 2.35138953418986841020647102354, 2.45945762114848560825649365176, 2.51670352312665363395238879815, 2.63269229092988112320683725222, 2.88680937668118379592671039789, 2.96577581991373207098789945165, 3.04164116748702008826977240626, 3.36487174547964856459707515691, 3.40389665926964421697743674206, 3.48512638490808730413728705295, 3.60518447905527237955920397297, 3.79161089902089641625625424940, 3.92670879058723171195278808026, 4.07683681254580985032185196946, 4.28736615428135818115492047015
Plot not available for L-functions of degree greater than 10.
