L(s) = 1 | + 13.4i·7-s − 17.6i·11-s − 7.48i·13-s + 16.9·17-s + 10.9·19-s + 21.9·23-s + 47.3i·29-s − 16.9·31-s + 5.53i·37-s + 66.3i·41-s + 38.9i·43-s + 32.5·47-s − 132.·49-s − 11.2·53-s + 31.8i·59-s + ⋯ |
L(s) = 1 | + 1.92i·7-s − 1.60i·11-s − 0.575i·13-s + 0.998·17-s + 0.577·19-s + 0.952·23-s + 1.63i·29-s − 0.547·31-s + 0.149i·37-s + 1.61i·41-s + 0.906i·43-s + 0.692·47-s − 2.71·49-s − 0.212·53-s + 0.540i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.831361870\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831361870\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 13.4iT - 49T^{2} \) |
| 11 | \( 1 + 17.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.48iT - 169T^{2} \) |
| 17 | \( 1 - 16.9T + 289T^{2} \) |
| 19 | \( 1 - 10.9T + 361T^{2} \) |
| 23 | \( 1 - 21.9T + 529T^{2} \) |
| 29 | \( 1 - 47.3iT - 841T^{2} \) |
| 31 | \( 1 + 16.9T + 961T^{2} \) |
| 37 | \( 1 - 5.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 38.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 31.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 77.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.92T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124. iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.948536142836437029479298143967, −9.012444583372463938280043908927, −8.567832478740728259403989429381, −7.70893479516687200027442009764, −6.38925815410310444150523399433, −5.58358196355765230664271984511, −5.15913593368000750720608303644, −3.28661728117934919873632028502, −2.83086382875173089209522586443, −1.18216442679400309577407966697,
0.69752858211954094291631151459, 1.95243950326073228082822925611, 3.58668376616940826839736702601, 4.29606008950813826360699526794, 5.18848535456667884524312540268, 6.59285531182819684585554879817, 7.40584386115651204901253028106, 7.63651898684938318757501550784, 9.130302288893371394135397985539, 9.948657741545333624493624501369