| L(s) = 1 | − 8·61-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 − 256-s + 257-s + ⋯ |
| L(s) = 1 | − 8·61-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 − 256-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2758139037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2758139037\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 + T^{8} \) |
| 5 | \( 1 + T^{8} \) |
| good | 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 23 | \( ( 1 + T^{8} )^{2} \) |
| 29 | \( ( 1 + T^{8} )^{2} \) |
| 31 | \( ( 1 + T^{4} )^{4} \) |
| 37 | \( ( 1 + T^{8} )^{2} \) |
| 41 | \( ( 1 + T^{4} )^{4} \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{8} )^{2} \) |
| 53 | \( ( 1 + T^{8} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
| 67 | \( ( 1 + T^{8} )^{2} \) |
| 71 | \( ( 1 + T^{4} )^{4} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−5.09470032567936546611171530725, −4.89092744545772404709306536763, −4.80245617053630373022698853134, −4.78325592094507981535842961147, −4.58487114272788382918758205339, −4.48100098493923577209838229287, −4.19915623439102622810059681238, −4.17024161667222865536322127642, −3.91637922139768922784361732789, −3.82138572584049933431629833425, −3.70002874252219234852205007549, −3.59973405390079576639375319791, −3.26770646524728254107296722957, −3.05845681824798920101211019981, −3.02051181319381739582656508481, −2.94753161345771772689683757979, −2.71289120383087434687676791588, −2.56683934784174094325973886809, −2.38601920806306834565530983533, −2.10434442424018758657396409286, −1.77080149460528092086803878449, −1.68204995784305245105306313746, −1.47872364347631136695807171386, −1.37374506817259377253998358129, −0.947679843246984856948564571077,
0.947679843246984856948564571077, 1.37374506817259377253998358129, 1.47872364347631136695807171386, 1.68204995784305245105306313746, 1.77080149460528092086803878449, 2.10434442424018758657396409286, 2.38601920806306834565530983533, 2.56683934784174094325973886809, 2.71289120383087434687676791588, 2.94753161345771772689683757979, 3.02051181319381739582656508481, 3.05845681824798920101211019981, 3.26770646524728254107296722957, 3.59973405390079576639375319791, 3.70002874252219234852205007549, 3.82138572584049933431629833425, 3.91637922139768922784361732789, 4.17024161667222865536322127642, 4.19915623439102622810059681238, 4.48100098493923577209838229287, 4.58487114272788382918758205339, 4.78325592094507981535842961147, 4.80245617053630373022698853134, 4.89092744545772404709306536763, 5.09470032567936546611171530725
Plot not available for L-functions of degree greater than 10.
