L(s) = 1 | + 4·13-s − 2·16-s + 4·37-s − 4·73-s + 8·97-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | + 4·13-s − 2·16-s + 4·37-s − 4·73-s + 8·97-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.458477665\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458477665\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T^{4} + T^{8} \) |
good | 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{2} )^{8} \) |
| 13 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | \( ( 1 + T^{4} )^{4} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{2} )^{8} \) |
| 73 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | \( ( 1 + T^{2} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \) |
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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
−4.09691269347198747243017800293, −4.08679409073861427608742218626, −4.08560928084775992775442789835, −3.98052729257964996945086056410, −3.63076968828590932984229228565, −3.61891125075105146221780404912, −3.51859965101872483672066882489, −3.25154234976711252282617464379, −3.23260049113057863805176944408, −3.09920885971472678980084070816, −2.99128388078274471816319378838, −2.79422779799053307114214757464, −2.68459359348795849812777953675, −2.47513300610500131550409083190, −2.34521297321712782304298575008, −2.19390629362508981274093439500, −2.13995327480068837656044663596, −1.99855369129168967017989672494, −1.79632143022749570785382123018, −1.37177126710068462463366644803, −1.31133797156050458918681170235, −1.25624921452704045376982932348, −1.15854273303398251252109637878, −0.956148891287532981728575777598, −0.54595665718497312685179890494,
0.54595665718497312685179890494, 0.956148891287532981728575777598, 1.15854273303398251252109637878, 1.25624921452704045376982932348, 1.31133797156050458918681170235, 1.37177126710068462463366644803, 1.79632143022749570785382123018, 1.99855369129168967017989672494, 2.13995327480068837656044663596, 2.19390629362508981274093439500, 2.34521297321712782304298575008, 2.47513300610500131550409083190, 2.68459359348795849812777953675, 2.79422779799053307114214757464, 2.99128388078274471816319378838, 3.09920885971472678980084070816, 3.23260049113057863805176944408, 3.25154234976711252282617464379, 3.51859965101872483672066882489, 3.61891125075105146221780404912, 3.63076968828590932984229228565, 3.98052729257964996945086056410, 4.08560928084775992775442789835, 4.08679409073861427608742218626, 4.09691269347198747243017800293
Plot not available for L-functions of degree greater than 10.