| L(s) = 1 | + (−1.85 + 1.45i)2-s + (−0.814 + 0.580i)3-s + (0.842 − 3.47i)4-s + (−0.200 − 0.0386i)5-s + (0.665 − 2.26i)6-s + (−0.836 + 2.50i)7-s + (1.54 + 3.38i)8-s + (0.327 − 0.945i)9-s + (0.428 − 0.221i)10-s + (−2.29 + 2.92i)11-s + (1.32 + 3.31i)12-s + (1.47 − 2.29i)13-s + (−2.10 − 5.87i)14-s + (0.185 − 0.0848i)15-s + (−1.44 − 0.744i)16-s + (−4.66 + 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 + 1.03i)2-s + (−0.470 + 0.334i)3-s + (0.421 − 1.73i)4-s + (−0.0897 − 0.0172i)5-s + (0.271 − 0.924i)6-s + (−0.316 + 0.948i)7-s + (0.545 + 1.19i)8-s + (0.109 − 0.315i)9-s + (0.135 − 0.0699i)10-s + (−0.692 + 0.881i)11-s + (0.383 + 0.957i)12-s + (0.409 − 0.636i)13-s + (−0.563 − 1.57i)14-s + (0.0479 − 0.0219i)15-s + (−0.361 − 0.186i)16-s + (−1.13 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0343416 - 0.0236795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0343416 - 0.0236795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 - 0.580i)T \) |
| 7 | \( 1 + (0.836 - 2.50i)T \) |
| 23 | \( 1 + (-3.55 + 3.21i)T \) |
| good | 2 | \( 1 + (1.85 - 1.45i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (0.200 + 0.0386i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (2.29 - 2.92i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.47 + 2.29i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.66 - 4.45i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.43 + 1.36i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (7.84 + 2.30i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.885 + 9.27i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.383 - 0.132i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 1.06i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-9.22 - 4.21i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-11.1 + 6.43i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.55 - 0.169i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.57 + 3.05i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-0.390 + 0.547i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (0.833 - 2.08i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.27 + 8.87i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (8.89 + 2.15i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (14.9 + 0.711i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.573 - 0.661i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (7.27 - 0.694i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (5.22 - 6.02i)T + (-13.8 - 96.0i)T^{2} \) |
| 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
−10.55160011275541257382115985487, −9.690213927832683153537998021594, −8.994264103013647410680588949138, −8.161812963557849097039881360374, −7.28733837393350491816090864769, −6.14723052216957896499094517836, −5.70256078041389000472250590507, −4.27911952228443995265593015064, −2.23898545233595424410734169747, −0.04297572529887053457773311605,
1.33623194548463097030420843993, 2.80747574411507195781418989867, 4.02052364711846835565480692223, 5.61451360517600063925800669358, 7.03095489186704260458762312755, 7.57769760174517901333084992953, 8.775305828816147791182108727307, 9.380623693611884483188014722366, 10.53159018782677565665524730411, 11.00674006943328698961586114629
