| L(s) = 1 | − 1.84·3-s + 10.5i·7-s − 5.60·9-s + 12.7·11-s − 23.0·13-s − 4.77i·17-s + (−1.66 + 18.9i)19-s − 19.4i·21-s + 35.7i·23-s + 26.9·27-s + 13.7i·29-s − 30.5i·31-s − 23.4·33-s − 54.8·37-s + 42.4·39-s + ⋯ |
| L(s) = 1 | − 0.614·3-s + 1.50i·7-s − 0.622·9-s + 1.15·11-s − 1.77·13-s − 0.280i·17-s + (−0.0878 + 0.996i)19-s − 0.924i·21-s + 1.55i·23-s + 0.996·27-s + 0.474i·29-s − 0.984i·31-s − 0.710·33-s − 1.48·37-s + 1.08·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06054269615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06054269615\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.66 - 18.9i)T \) |
| good | 3 | \( 1 + 1.84T + 9T^{2} \) |
| 7 | \( 1 - 10.5iT - 49T^{2} \) |
| 11 | \( 1 - 12.7T + 121T^{2} \) |
| 13 | \( 1 + 23.0T + 169T^{2} \) |
| 17 | \( 1 + 4.77iT - 289T^{2} \) |
| 23 | \( 1 - 35.7iT - 529T^{2} \) |
| 29 | \( 1 - 13.7iT - 841T^{2} \) |
| 31 | \( 1 + 30.5iT - 961T^{2} \) |
| 37 | \( 1 + 54.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 10.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 54.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.95iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 130. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 71.2T + 9.40e3T^{2} \) |
| 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
−9.470381867108455751949483787563, −8.916078978370453684938977950950, −8.030806767589523972269143555032, −7.09022756710671800184462760704, −6.23498465168989589796334024708, −5.46444787734544400570155208758, −5.05435328235068549373075600067, −3.71037528322208831786160001180, −2.66083288130079900628704822734, −1.70779511533655301186437399486,
0.02041616664769086447553795344, 0.911587672693289381851101203289, 2.38760320891475268604261767957, 3.56077794954544057899789402446, 4.56803613917991985607238411915, 5.03580021440709151618045946640, 6.36898152816902536737561939444, 6.84155125413868062492114062503, 7.51077404977486396918330622062, 8.575796495466365109639511597961
