L(s) = 1 | + 9-s − 2·17-s − 25-s − 2·41-s + 49-s + 2·73-s + 81-s + 2·89-s − 2·97-s + 2·113-s + ⋯ |
L(s) = 1 | + 9-s − 2·17-s − 25-s − 2·41-s + 49-s + 2·73-s + 81-s + 2·89-s − 2·97-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7559661027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7559661027\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.26505903510516084641264694255, −11.27425088452062960443173901726, −10.35217709081203788520341877720, −9.411931474298749010055050808183, −8.412844904341925196918485700193, −7.21695810024518924745425559826, −6.38107797641790162398645317460, −4.90430441146662750205657053043, −3.86766998712323449552352878936, −2.05854479506742573974032235914,
2.05854479506742573974032235914, 3.86766998712323449552352878936, 4.90430441146662750205657053043, 6.38107797641790162398645317460, 7.21695810024518924745425559826, 8.412844904341925196918485700193, 9.411931474298749010055050808183, 10.35217709081203788520341877720, 11.27425088452062960443173901726, 12.26505903510516084641264694255