| L(s) = 1 | + (−0.258 − 0.216i)2-s + (0.356 − 0.617i)3-s + (−0.327 − 1.85i)4-s + (−1.04 − 1.24i)5-s + (−0.225 + 0.0821i)6-s + (2.10 + 1.21i)7-s + (−0.655 + 1.13i)8-s + (1.24 + 2.15i)9-s + 0.546i·10-s + (−0.682 − 0.813i)11-s + (−1.26 − 0.459i)12-s + (0.425 + 1.16i)13-s + (−0.280 − 0.770i)14-s + (−1.13 + 0.200i)15-s + (−3.13 + 1.13i)16-s + (−0.818 + 0.472i)17-s + ⋯ |
| L(s) = 1 | + (−0.182 − 0.153i)2-s + (0.205 − 0.356i)3-s + (−0.163 − 0.928i)4-s + (−0.466 − 0.555i)5-s + (−0.0921 + 0.0335i)6-s + (0.795 + 0.459i)7-s + (−0.231 + 0.401i)8-s + (0.415 + 0.719i)9-s + 0.172i·10-s + (−0.205 − 0.245i)11-s + (−0.364 − 0.132i)12-s + (0.117 + 0.324i)13-s + (−0.0749 − 0.205i)14-s + (−0.294 + 0.0518i)15-s + (−0.782 + 0.284i)16-s + (−0.198 + 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.762745 - 0.418755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.762745 - 0.418755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (-8.53 + 0.262i)T \) |
| good | 2 | \( 1 + (0.258 + 0.216i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.356 + 0.617i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.04 + 1.24i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.10 - 1.21i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.682 + 0.813i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.425 - 1.16i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.818 - 0.472i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.03 - 2.54i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.37 - 4.51i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.08 + 3.67i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.95 + 0.873i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.54 + 2.97i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (11.4 + 4.15i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-7.15 - 4.13i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.07 - 5.69i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.21 + 2.63i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.69 - 4.66i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (12.9 + 4.72i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (9.25 - 3.36i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (1.17 + 0.987i)T + (12.3 + 69.9i)T^{2} \) |
| 79 | \( 1 + (8.25 - 3.00i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + 8.42iT - 83T^{2} \) |
| 89 | \( 1 + (0.119 - 0.0436i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.66 - 4.61i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−14.31388657139288414616111662680, −13.54293997826837191357106893339, −12.13572745904189947349310119227, −11.15795938800732052536452319147, −9.944873514493525905116967768014, −8.635482061642704819517839403581, −7.69800377247524656615135625707, −5.79570095267181107414575169850, −4.57724690432587544107081231783, −1.79127704656411945883154325886,
3.30469638809925891314143186394, 4.54371680990307077689560334977, 6.87678939602643512064534268872, 7.80491402705472144509452476268, 8.953451709858649997544757535220, 10.31640099287008377079699219855, 11.52709677127676911971636805849, 12.52756150646010879103364408258, 13.80236647115812895392157526848, 14.94025034375311745491282858249
