L(s) = 1 | + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-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 | + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.439588194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.439588194\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 3 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 29 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 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 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 71 | \( ( 1 - T + T^{2} )^{8} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 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 | \( ( 1 + T^{4} )^{4} \) |
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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
−4.36941610548638409971685415239, −4.35851928887375527034853727680, −3.96647107914123006314442429121, −3.96349469404813819988081851981, −3.86968123972594648253967970894, −3.84273784607651024876426960417, −3.71606341840407907275218434847, −3.41650230896378843816209700367, −3.31354517513423685009325684271, −3.29867726749598220788928904031, −3.28801632691664035382696438772, −3.06351547788081039922688455579, −2.89298250954082879827283951574, −2.46475113049638324653093542350, −2.31278192804378191283822101731, −2.22007634335971687663320888564, −2.17338782806571675765020096539, −2.11991578274121113102865490538, −2.02665743597294185186957471526, −1.80517201745387408000729403296, −1.27619977569278877705028896279, −1.19578509012443993254680863876, −1.14208435777253230841162635923, −1.06208500442160328966174418076, −0.795073279066333925922302132925,
0.795073279066333925922302132925, 1.06208500442160328966174418076, 1.14208435777253230841162635923, 1.19578509012443993254680863876, 1.27619977569278877705028896279, 1.80517201745387408000729403296, 2.02665743597294185186957471526, 2.11991578274121113102865490538, 2.17338782806571675765020096539, 2.22007634335971687663320888564, 2.31278192804378191283822101731, 2.46475113049638324653093542350, 2.89298250954082879827283951574, 3.06351547788081039922688455579, 3.28801632691664035382696438772, 3.29867726749598220788928904031, 3.31354517513423685009325684271, 3.41650230896378843816209700367, 3.71606341840407907275218434847, 3.84273784607651024876426960417, 3.86968123972594648253967970894, 3.96349469404813819988081851981, 3.96647107914123006314442429121, 4.35851928887375527034853727680, 4.36941610548638409971685415239
Plot not available for L-functions of degree greater than 10.