L(s) = 1 | − 3.29i·5-s + 2.46·7-s − 10.4i·11-s + 2.93·13-s − 27.8i·17-s + 4.35·19-s + 24.3i·23-s + 14.1·25-s − 7.80i·29-s + 17.4·31-s − 8.11i·35-s − 48.3·37-s − 51.2i·41-s + 82.7·43-s − 22.2i·47-s + ⋯ |
L(s) = 1 | − 0.659i·5-s + 0.351·7-s − 0.946i·11-s + 0.225·13-s − 1.63i·17-s + 0.229·19-s + 1.05i·23-s + 0.565·25-s − 0.269i·29-s + 0.562·31-s − 0.231i·35-s − 1.30·37-s − 1.24i·41-s + 1.92·43-s − 0.472i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.780486314\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780486314\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
good | 5 | \( 1 + 3.29iT - 25T^{2} \) |
| 7 | \( 1 - 2.46T + 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 2.93T + 169T^{2} \) |
| 17 | \( 1 + 27.8iT - 289T^{2} \) |
| 23 | \( 1 - 24.3iT - 529T^{2} \) |
| 29 | \( 1 + 7.80iT - 841T^{2} \) |
| 31 | \( 1 - 17.4T + 961T^{2} \) |
| 37 | \( 1 + 48.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 51.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 22.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 49.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 1.05T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 137.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 4.56iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13.4T + 9.40e3T^{2} \) |
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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
−8.632463550747653659190631117259, −7.57444252305955207012695023359, −7.04190831932895351534421579641, −5.86343970157321095850339897543, −5.31202975902303444637409665254, −4.54147376784671432730810861688, −3.52454116835914242439740401617, −2.63543127509638403146209030017, −1.33120192575877164837269212714, −0.43745811850339143720514683145,
1.28747003470351549491335247565, 2.25589110281936665105561478224, 3.23056894493819671521304726178, 4.22111419741005675068607742078, 4.89641188305264077382649163621, 6.02765167841203906511483982829, 6.59409324587649368894467056396, 7.38668975346161901153290623023, 8.150747362843536545792638706202, 8.808305193158162219345170729181