L(s) = 1 | + (−0.866 − 0.629i)2-s + (0.0458 + 0.141i)4-s + (−1.41 + 1.03i)5-s + (−0.282 + 0.867i)8-s + (−0.809 − 0.587i)9-s + 1.87·10-s + (0.535 − 0.844i)11-s + (1.03 + 0.749i)13-s + (0.911 − 0.662i)16-s + (0.331 + 1.01i)18-s + (0.450 − 1.38i)19-s + (−0.210 − 0.152i)20-s + (−0.996 + 0.394i)22-s + 1.45·23-s + (0.640 − 1.97i)25-s + (−0.422 − 1.29i)26-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.629i)2-s + (0.0458 + 0.141i)4-s + (−1.41 + 1.03i)5-s + (−0.282 + 0.867i)8-s + (−0.809 − 0.587i)9-s + 1.87·10-s + (0.535 − 0.844i)11-s + (1.03 + 0.749i)13-s + (0.911 − 0.662i)16-s + (0.331 + 1.01i)18-s + (0.450 − 1.38i)19-s + (−0.210 − 0.152i)20-s + (−0.996 + 0.394i)22-s + 1.45·23-s + (0.640 − 1.97i)25-s + (−0.422 − 1.29i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4709761521\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4709761521\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.535 + 0.844i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - 1.45T + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 0.374T + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 1.93T + T^{2} \) |
| 97 | \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.55686292839872533604222653086, −9.137214106156333698445876358627, −8.829024836155047620224899820932, −7.991614813591565222193887346137, −6.85223172072686921557647690387, −6.21877177800301261599284433453, −4.74419491286681973010414489617, −3.37181503313948160286045968467, −2.91928111320206341655274156436, −0.889662271639472934932589881810,
1.03889774240273580466912475836, 3.30365295213831375178309489017, 4.16406764218319890375043261541, 5.22546651144970450619600551112, 6.37075458247018337521396068690, 7.52865329453520918882018071847, 8.032686944840645017517574594755, 8.539641499652409876233935021862, 9.279969913525273664350950539070, 10.28525766066797671766918245685