| L(s) = 1 | + 0.508i·2-s + 1.74·4-s + 1.10·5-s + 1.90i·8-s + 3·9-s + 0.561i·10-s + (1.60 + 2.90i)11-s − 6.81i·13-s + 2.51·16-s + 1.52i·18-s + 1.50i·19-s + 1.92·20-s + (−1.47 + 0.817i)22-s − 6.37·23-s − 3.78·25-s + 3.46·26-s + ⋯ |
| L(s) = 1 | + 0.359i·2-s + 0.870·4-s + 0.493·5-s + 0.672i·8-s + 9-s + 0.177i·10-s + (0.484 + 0.874i)11-s − 1.89i·13-s + 0.628·16-s + 0.359i·18-s + 0.345i·19-s + 0.429·20-s + (−0.314 + 0.174i)22-s − 1.33·23-s − 0.756·25-s + 0.679·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23374 + 0.577817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23374 + 0.577817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.60 - 2.90i)T \) |
| 79 | \( 1 - 8.88iT \) |
| good | 2 | \( 1 - 0.508iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 6.81iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 1.50iT - 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7.91T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4.17T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.99iT - 73T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.08T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−10.04051106623832004153111912819, −9.777142860562419518954833609750, −8.146929719139973077478035126890, −7.73421533142939129293250234036, −6.72417479846262112023037016180, −6.02456605623017537039379663099, −5.11348006781686936262212704976, −3.86362157605809422974597389372, −2.54963563677362845324830633453, −1.47287777088491741198450900283,
1.42334775410312292715171023487, 2.25759378336277151197963656056, 3.66456628948844002650249352532, 4.50763450813237735109210699789, 6.07143021719477969228191657688, 6.52117471488159043035966735479, 7.37024560447791965715826003728, 8.466109593898758599177111298281, 9.579195606111945114783487620973, 9.975722744437064922279812859966
