L(s) = 1 | − 3.89·3-s + 8.85·5-s + 6.18·9-s − 3.64i·11-s − 7.79·13-s − 34.5·15-s + 10.4i·17-s + 10.7·19-s + 12.9·23-s + 53.4·25-s + 10.9·27-s + 17.2i·29-s − 30.2i·31-s + 14.2i·33-s + 39.5i·37-s + ⋯ |
L(s) = 1 | − 1.29·3-s + 1.77·5-s + 0.686·9-s − 0.331i·11-s − 0.599·13-s − 2.30·15-s + 0.616i·17-s + 0.567·19-s + 0.561·23-s + 2.13·25-s + 0.406·27-s + 0.594i·29-s − 0.975i·31-s + 0.430i·33-s + 1.06i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.710225287\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710225287\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 3.89T + 9T^{2} \) |
| 5 | \( 1 - 8.85T + 25T^{2} \) |
| 11 | \( 1 + 3.64iT - 121T^{2} \) |
| 13 | \( 1 + 7.79T + 169T^{2} \) |
| 17 | \( 1 - 10.4iT - 289T^{2} \) |
| 19 | \( 1 - 10.7T + 361T^{2} \) |
| 23 | \( 1 - 12.9T + 529T^{2} \) |
| 29 | \( 1 - 17.2iT - 841T^{2} \) |
| 31 | \( 1 + 30.2iT - 961T^{2} \) |
| 37 | \( 1 - 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.41iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 15.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 7.79iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 89.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 70.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 27.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.26iT - 9.40e3T^{2} \) |
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
−9.467458693384469223865360884291, −8.689461426831939192134523612439, −7.41345569637786755744046513677, −6.48206018327061248888302779055, −5.96984531068307720057511167336, −5.34053119912415108258282530148, −4.66662435448856250915634258783, −3.06360434136942807216396789383, −1.92618605548121408009007987716, −0.847944439779846908601984398279,
0.76010118436500014015642837476, 1.91585845838343473366189104178, 2.94722771683891204571960221722, 4.66442475180012533846728452856, 5.28530433298807928439290829648, 5.81327903440790366414351424224, 6.66252916718926267940459709494, 7.22838163938605732328171245088, 8.629860500077863321380425544486, 9.566939130616614490870692881383