| L(s) = 1 | + (0.483 − 1.32i)2-s + (−1.53 − 1.28i)4-s + (−3.98 − 0.702i)5-s + (−10.1 + 8.49i)7-s + (−2.44 + 1.41i)8-s + (−2.86 + 4.95i)10-s + (−13.3 + 2.35i)11-s + (17.4 − 6.34i)13-s + (6.38 + 17.5i)14-s + (0.694 + 3.93i)16-s + (−13.8 − 8.01i)17-s + (0.327 + 0.566i)19-s + (5.20 + 6.20i)20-s + (−3.33 + 18.9i)22-s + (−4.24 + 5.05i)23-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.383 − 0.321i)4-s + (−0.797 − 0.140i)5-s + (−1.44 + 1.21i)7-s + (−0.306 + 0.176i)8-s + (−0.286 + 0.495i)10-s + (−1.21 + 0.214i)11-s + (1.34 − 0.488i)13-s + (0.456 + 1.25i)14-s + (0.0434 + 0.246i)16-s + (−0.816 − 0.471i)17-s + (0.0172 + 0.0298i)19-s + (0.260 + 0.310i)20-s + (−0.151 + 0.859i)22-s + (−0.184 + 0.219i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0372299 + 0.0824702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0372299 + 0.0824702i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 + 1.32i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.98 + 0.702i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (10.1 - 8.49i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (13.3 - 2.35i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 6.34i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (13.8 + 8.01i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-0.327 - 0.566i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.24 - 5.05i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (0.466 - 1.28i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 12.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (8.43 - 14.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-14.6 - 40.2i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-0.0113 - 0.0645i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (30.0 + 35.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 14.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-4.79 - 0.844i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (12.0 - 10.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (37.1 - 13.5i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-60.9 - 35.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (34.1 + 59.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-47.4 - 17.2i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (50.2 - 138. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-17.2 + 9.98i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-18.3 - 104. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
| 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
−12.98995957667066063390696880687, −12.05385520907297935765148382031, −11.16136427445483697624745708575, −10.04045616457782421040656066602, −9.026350802608260599507365960124, −8.049849890259000524225528704631, −6.39772820778225540445057766776, −5.31827671199448563898875205943, −3.70060600365759261015175394778, −2.61943602446652544078534752106,
0.04950565642678789414651133636, 3.38007832431437299686186697445, 4.21896557917352071691203968743, 6.00247786773452966510476777116, 6.94146251782382411763744548253, 7.85944245198550259716625852336, 9.015005583855484722384970758077, 10.35642351786452880798456278981, 11.12775229618260989455191113862, 12.66548869033881498015942344827
