L(s) = 1 | + (−1.91 + 0.252i)3-s + (0.258 + 0.965i)4-s + (−0.583 − 0.665i)5-s + (2.63 − 0.707i)9-s + (0.608 + 0.793i)11-s + (−0.739 − 1.78i)12-s + (1.28 + 1.12i)15-s + (−0.866 + 0.499i)16-s + (0.491 − 0.735i)20-s + (−1.34 + 0.665i)23-s + (0.0283 − 0.215i)25-s + (−3.09 + 1.28i)27-s + (0.241 + 1.83i)31-s + (−1.36 − 1.36i)33-s + (1.36 + 2.36i)36-s + (−0.837 + 1.69i)37-s + ⋯ |
L(s) = 1 | + (−1.91 + 0.252i)3-s + (0.258 + 0.965i)4-s + (−0.583 − 0.665i)5-s + (2.63 − 0.707i)9-s + (0.608 + 0.793i)11-s + (−0.739 − 1.78i)12-s + (1.28 + 1.12i)15-s + (−0.866 + 0.499i)16-s + (0.491 − 0.735i)20-s + (−1.34 + 0.665i)23-s + (0.0283 − 0.215i)25-s + (−3.09 + 1.28i)27-s + (0.241 + 1.83i)31-s + (−1.36 − 1.36i)33-s + (1.36 + 2.36i)36-s + (−0.837 + 1.69i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3983532344\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3983532344\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.608 - 0.793i)T \) |
| 97 | \( 1 + (-0.793 - 0.608i)T \) |
good | 2 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 3 | \( 1 + (1.91 - 0.252i)T + (0.965 - 0.258i)T^{2} \) |
| 5 | \( 1 + (0.583 + 0.665i)T + (-0.130 + 0.991i)T^{2} \) |
| 7 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 13 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 17 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 19 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (1.34 - 0.665i)T + (0.608 - 0.793i)T^{2} \) |
| 29 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 31 | \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \) |
| 37 | \( 1 + (0.837 - 1.69i)T + (-0.608 - 0.793i)T^{2} \) |
| 41 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (0.158 - 0.207i)T + (-0.258 - 0.965i)T^{2} \) |
| 59 | \( 1 + (1.60 + 0.793i)T + (0.608 + 0.793i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.389 - 1.95i)T + (-0.923 + 0.382i)T^{2} \) |
| 71 | \( 1 + (-0.641 + 0.0420i)T + (0.991 - 0.130i)T^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 89 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
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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
−10.48970832356248602723827029798, −9.779598415177131522765325394831, −8.683028722100993202608863378925, −7.69767622272313164233063121785, −6.87931395923671393649565159795, −6.25363408755655974220062030252, −5.02989682934968707059142222053, −4.45064811805642389879154917211, −3.62335984153133250977262623178, −1.52898247956457764245526933805,
0.48722567172832297592643217364, 1.93418041285458702402076523709, 3.85927334219657456564248780013, 4.80130603105633559417745539836, 5.97370126188045452606222540556, 6.08224663551071017512432328653, 7.02229543423569831500577936472, 7.77269269607911768122853085310, 9.323421236769868842396838820479, 10.18850922347195276833761163095