| L(s) = 1 | + 8·13-s + 40·37-s − 56·61-s − 8·109-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 + 239-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 6.57·37-s − 7.17·61-s − 0.766·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182008024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182008024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 2 T + p T^{2} )^{4}( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 46 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 194 T^{4} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 14 T + p T^{2} )^{4}( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | \( ( 1 + 5906 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 + 7682 T^{4} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.76989677159229344038473931681, −3.66243766781013881886716977819, −3.48698054325897862424510666638, −3.39612916553051431304691875738, −3.18219709544077357349458994975, −3.10562037065779567244079280093, −3.08438486079658086701868200140, −2.99020854895387734454766143423, −2.77142668409070737687800926766, −2.62214359484035452841674696322, −2.58787161712220285390004411847, −2.44610968035787209939317576470, −2.27753846910265299087436406223, −2.23835924680964445355596804284, −1.74239218752982577622006665020, −1.67061549212658918453612669832, −1.63357317184786226376672506814, −1.62177224685869915673908173932, −1.49240085889563123572732993797, −1.12497876184753900303032513162, −0.888487536748921548333230445150, −0.822868703384004403576106687549, −0.808282490628075205722926757131, −0.51284004097641588083151319081, −0.07244231811309053405953560284,
0.07244231811309053405953560284, 0.51284004097641588083151319081, 0.808282490628075205722926757131, 0.822868703384004403576106687549, 0.888487536748921548333230445150, 1.12497876184753900303032513162, 1.49240085889563123572732993797, 1.62177224685869915673908173932, 1.63357317184786226376672506814, 1.67061549212658918453612669832, 1.74239218752982577622006665020, 2.23835924680964445355596804284, 2.27753846910265299087436406223, 2.44610968035787209939317576470, 2.58787161712220285390004411847, 2.62214359484035452841674696322, 2.77142668409070737687800926766, 2.99020854895387734454766143423, 3.08438486079658086701868200140, 3.10562037065779567244079280093, 3.18219709544077357349458994975, 3.39612916553051431304691875738, 3.48698054325897862424510666638, 3.66243766781013881886716977819, 3.76989677159229344038473931681
Plot not available for L-functions of degree greater than 10.
