| L(s) = 1 | + (−0.647 + 1.87i)2-s + (0.458 + 0.888i)3-s + (−1.51 − 1.18i)4-s + (−3.91 + 0.374i)5-s + (−1.95 + 0.281i)6-s + (−1.01 − 2.44i)7-s + (−0.131 + 0.0845i)8-s + (−0.580 + 0.814i)9-s + (1.83 − 7.57i)10-s + (0.645 − 0.223i)11-s + (0.363 − 1.88i)12-s + (−0.0976 − 0.332i)13-s + (5.22 − 0.326i)14-s + (−2.12 − 3.31i)15-s + (−0.978 − 4.03i)16-s + (5.85 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.457 + 1.32i)2-s + (0.264 + 0.513i)3-s + (−0.755 − 0.593i)4-s + (−1.75 + 0.167i)5-s + (−0.800 + 0.115i)6-s + (−0.385 − 0.922i)7-s + (−0.0465 + 0.0299i)8-s + (−0.193 + 0.271i)9-s + (0.581 − 2.39i)10-s + (0.194 − 0.0673i)11-s + (0.104 − 0.544i)12-s + (−0.0270 − 0.0922i)13-s + (1.39 − 0.0872i)14-s + (−0.549 − 0.855i)15-s + (−0.244 − 1.00i)16-s + (1.42 + 0.568i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.204108 - 0.0626517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204108 - 0.0626517i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (1.01 + 2.44i)T \) |
| 23 | \( 1 + (4.15 + 2.40i)T \) |
| good | 2 | \( 1 + (0.647 - 1.87i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (3.91 - 0.374i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.645 + 0.223i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.0976 + 0.332i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-5.85 - 2.34i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 0.646i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.951 + 6.61i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.91 - 0.233i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (8.88 + 6.32i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (10.0 + 4.58i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.14 + 3.33i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (6.49 - 3.74i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0565 - 0.0593i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (7.52 + 1.82i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 3.35i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (0.257 + 1.33i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-0.530 - 0.611i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.37 - 1.74i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-4.52 + 4.74i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-5.09 - 11.1i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.122 + 2.57i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (6.54 - 14.3i)T + (-63.5 - 73.3i)T^{2} \) |
| 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
−10.70608170638127199564172136747, −9.863096972800300595805184527381, −8.741497553987749582816860904593, −7.911846830916444354710078361341, −7.51338956249247834678093135241, −6.60831474622068597372462035076, −5.33279593146685864733608962358, −4.02747935120507816013579353000, −3.39038971949446792811887449534, −0.15107377400582078046513207736,
1.51375134635917852585799959433, 3.17877649324743354382070619068, 3.54206938507907269719791250968, 5.18111920749907302202300400625, 6.70482101541062562727013229406, 7.76974050998923409204020869845, 8.506131083259939484402620485577, 9.313966164524659139275914224001, 10.23440689199988599450326993133, 11.43085647434694962212773956569
