L(s) = 1 | + (−1.86 + 1.23i)5-s + 0.746i·7-s − 5.36i·11-s + 2.92·13-s + 2.13i·17-s − 1.73i·19-s + 7.49i·23-s + (1.92 − 4.61i)25-s − 6.74i·29-s − 2.64·31-s + (−0.925 − 1.38i)35-s + 1.07·37-s + 11.2·41-s + 7.44·43-s − 1.73i·47-s + ⋯ |
L(s) = 1 | + (−0.832 + 0.554i)5-s + 0.282i·7-s − 1.61i·11-s + 0.811·13-s + 0.517i·17-s − 0.397i·19-s + 1.56i·23-s + (0.385 − 0.922i)25-s − 1.25i·29-s − 0.475·31-s + (−0.156 − 0.234i)35-s + 0.176·37-s + 1.76·41-s + 1.13·43-s − 0.252i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.422934077\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.422934077\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
good | 7 | \( 1 - 0.746iT - 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.49iT - 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.44T + 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 - 7.72T + 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 - 6.45iT - 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.690iT - 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−9.369108267664207845150223080798, −8.659440573750684398000480554052, −7.941818982631089650395871797613, −7.22988677271648971729368854097, −6.04777663626052484510226028914, −5.70262920222706886457064396332, −4.14360400439379396517105803277, −3.54867030864695558088172142375, −2.56234823153899723586485100445, −0.823918412908044741714967297996,
0.922303307040343769820116804871, 2.31130988366046038856588739131, 3.70960466473168869672699744930, 4.41979976999778651062362316520, 5.15194923687295867972652730513, 6.38421829250710245083018921457, 7.27658283067576664508501959727, 7.80588758594720260577122228134, 8.810832452149811398883041375420, 9.358194987342478183705101645354