L(s) = 1 | − 2·16-s + 8·23-s − 8·121-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 + 251-s + ⋯ |
L(s) = 1 | − 2·16-s + 8·23-s − 8·121-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 + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7126950702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7126950702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 + T^{8} \) |
| 5 | \( 1 + T^{8} \) |
| 7 | \( ( 1 + T^{4} )^{2} \) |
good | 11 | \( ( 1 + T^{2} )^{8} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{8} )^{2} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{4} )^{4} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 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
−4.77378272014413697342364915032, −4.57950626846285614279986335162, −4.42476091110298767738284224309, −4.41911498370607135580904484895, −4.24286165532814368269559004282, −3.92403855270776471374600829562, −3.88125212930056334834536621952, −3.54891342543607918975105032350, −3.50255551713880824964302553645, −3.39360023381997439146620599089, −3.30468343959032288589146709290, −3.29815835664180795271384971436, −2.98122520361541968056693347761, −2.63594761260926485056462840590, −2.61627501555961656032148228511, −2.55766681738740449173356675052, −2.55395093085680873567685738690, −2.43594433320214708286483971695, −2.17313963422584616178933671452, −1.75471889459808690040085823476, −1.50985470321534075997042451947, −1.32865907606311989041910774252, −1.26079521219839814510303856886, −1.04836977250753396050973636224, −0.872934332713651999777288495447,
0.872934332713651999777288495447, 1.04836977250753396050973636224, 1.26079521219839814510303856886, 1.32865907606311989041910774252, 1.50985470321534075997042451947, 1.75471889459808690040085823476, 2.17313963422584616178933671452, 2.43594433320214708286483971695, 2.55395093085680873567685738690, 2.55766681738740449173356675052, 2.61627501555961656032148228511, 2.63594761260926485056462840590, 2.98122520361541968056693347761, 3.29815835664180795271384971436, 3.30468343959032288589146709290, 3.39360023381997439146620599089, 3.50255551713880824964302553645, 3.54891342543607918975105032350, 3.88125212930056334834536621952, 3.92403855270776471374600829562, 4.24286165532814368269559004282, 4.41911498370607135580904484895, 4.42476091110298767738284224309, 4.57950626846285614279986335162, 4.77378272014413697342364915032
Plot not available for L-functions of degree greater than 10.