L(s) = 1 | + (−2.49 − 1.67i)3-s + (4.19 + 2.71i)5-s − 12.7i·7-s + (3.41 + 8.32i)9-s + 12.6i·11-s + 7.44i·13-s + (−5.92 − 13.7i)15-s − 14.0·17-s − 31.0·19-s + (−21.3 + 31.8i)21-s − 7.50·23-s + (10.2 + 22.7i)25-s + (5.40 − 26.4i)27-s − 15.7i·29-s + 20.4·31-s + ⋯ |
L(s) = 1 | + (−0.830 − 0.557i)3-s + (0.839 + 0.542i)5-s − 1.82i·7-s + (0.379 + 0.925i)9-s + 1.14i·11-s + 0.572i·13-s + (−0.395 − 0.918i)15-s − 0.826·17-s − 1.63·19-s + (−1.01 + 1.51i)21-s − 0.326·23-s + (0.410 + 0.911i)25-s + (0.200 − 0.979i)27-s − 0.542i·29-s + 0.660·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5000988261\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5000988261\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.49 + 1.67i)T \) |
| 5 | \( 1 + (-4.19 - 2.71i)T \) |
good | 7 | \( 1 + 12.7iT - 49T^{2} \) |
| 11 | \( 1 - 12.6iT - 121T^{2} \) |
| 13 | \( 1 - 7.44iT - 169T^{2} \) |
| 17 | \( 1 + 14.0T + 289T^{2} \) |
| 19 | \( 1 + 31.0T + 361T^{2} \) |
| 23 | \( 1 + 7.50T + 529T^{2} \) |
| 29 | \( 1 + 15.7iT - 841T^{2} \) |
| 31 | \( 1 - 20.4T + 961T^{2} \) |
| 37 | \( 1 - 12.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 47.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 0.630iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 37.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 140. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 10.3iT - 9.40e3T^{2} \) |
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
−10.25466692129640022831076064994, −9.629915327558659444851441760127, −8.202308804253621132973359614194, −7.16447944398407709652712246892, −6.78268976905213918977933913218, −6.11369282413333776585145214982, −4.66912327474830731858831615803, −4.18196428031038875419981634522, −2.32801753666551503952383087842, −1.39440499926958857125972768990,
0.17221118211512462953765797920, 1.89937103141617184106188250887, 3.04464098029439324995748423622, 4.53103180362860124118403893626, 5.34475786184046217699200231959, 6.04859482426994309397951671541, 6.42440098330333236317367497531, 8.359476817038238962698687627021, 8.798423290516944800298032303219, 9.479307680613982299796240746414