L(s) = 1 | + 1.73·3-s − 1.36i·5-s + 1.24i·7-s + 2.99·9-s − 5.79·11-s + 16.3i·13-s − 2.36i·15-s − 5.01·17-s + 26.1·19-s + 2.15i·21-s + 25.1i·23-s + 23.1·25-s + 5.19·27-s + 32.7i·29-s + 1.01i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.272i·5-s + 0.177i·7-s + 0.333·9-s − 0.527·11-s + 1.26i·13-s − 0.157i·15-s − 0.294·17-s + 1.37·19-s + 0.102i·21-s + 1.09i·23-s + 0.925·25-s + 0.192·27-s + 1.13i·29-s + 0.0328i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.178223586\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.178223586\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 + 1.36iT - 25T^{2} \) |
| 7 | \( 1 - 1.24iT - 49T^{2} \) |
| 11 | \( 1 + 5.79T + 121T^{2} \) |
| 13 | \( 1 - 16.3iT - 169T^{2} \) |
| 17 | \( 1 + 5.01T + 289T^{2} \) |
| 19 | \( 1 - 26.1T + 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 - 32.7iT - 841T^{2} \) |
| 31 | \( 1 - 1.01iT - 961T^{2} \) |
| 37 | \( 1 + 14.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 20.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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
−10.08157658884069369034058818169, −9.168532462902816005065965144918, −8.774713638300959652997987225723, −7.55666137618480044871614741872, −7.00713020202512827178094445883, −5.69182130147791028771608919322, −4.78984255550321175362242367867, −3.69239314990055361126645869615, −2.58500709697493714937025706959, −1.31720028970895984879773677167,
0.77277943973742958823157742571, 2.51153292759672768766481950477, 3.25041601879863790242149026798, 4.51741853919127821831025142002, 5.52556637010224764537660065908, 6.60650851534584841660543298489, 7.64662412910167011140008984624, 8.132625848002627372413619995231, 9.195364571922283933530052168947, 10.05345094225084982132001179871