L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.419 − 2.37i)3-s + (0.766 + 0.642i)4-s + (0.419 − 2.37i)6-s + (−1.31 + 2.27i)7-s + (0.500 + 0.866i)8-s + (−2.65 + 0.968i)9-s + (−2.09 − 3.62i)11-s + (1.20 − 2.09i)12-s + (0.544 − 3.08i)13-s + (−2.01 + 1.68i)14-s + (0.173 + 0.984i)16-s + (−7.14 − 2.60i)17-s − 2.83·18-s + (2.25 − 3.72i)19-s + ⋯ |
L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.242 − 1.37i)3-s + (0.383 + 0.321i)4-s + (0.171 − 0.970i)6-s + (−0.496 + 0.860i)7-s + (0.176 + 0.306i)8-s + (−0.886 + 0.322i)9-s + (−0.631 − 1.09i)11-s + (0.348 − 0.603i)12-s + (0.150 − 0.855i)13-s + (−0.538 + 0.451i)14-s + (0.0434 + 0.246i)16-s + (−1.73 − 0.630i)17-s − 0.667·18-s + (0.517 − 0.855i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.557133 - 1.29356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.557133 - 1.29356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.25 + 3.72i)T \) |
good | 3 | \( 1 + (0.419 + 2.37i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (1.31 - 2.27i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.09 + 3.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.544 + 3.08i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (7.14 + 2.60i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.50 + 3.77i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.18 + 1.88i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.321 - 0.556i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + (-0.611 - 3.47i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.48 + 2.08i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.14 - 2.59i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (4.83 + 4.05i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.20 - 1.16i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.95 + 2.47i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.95 + 1.07i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.14 + 0.961i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.900 - 5.10i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.0835 - 0.474i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (5.16 - 8.94i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.65 - 15.0i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.20 - 2.62i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.650433386421456588659400742464, −8.492099166463006472693397722124, −8.012904441252239078687521365995, −6.91687558832805213262711414510, −6.27196154168207575006710943762, −5.69507691353236857044029820936, −4.59403132565615296395589154731, −2.93287928037535352822854588654, −2.39832915554042945299216086208, −0.50966404109123013310778159389,
1.98346385900419869833376460836, 3.45375146850327061124465194750, 4.30750264860844942515954857464, 4.63688769017572497831393155156, 5.86768107992071060877136823322, 6.75125712066558505643946293478, 7.71436357888923832996192117147, 9.046286732269650378912565368494, 9.901750462974960861894274750163, 10.25009948665909180121613409592