| L(s) = 1 | + (−8.04e4 − 4.64e4i)2-s + (4.22e7 + 8.40e6i)3-s + (2.16e9 + 3.75e9i)4-s + (1.68e11 + 9.71e10i)5-s + (−3.00e12 − 2.63e12i)6-s + (−2.65e13 − 1.99e13i)7-s + (0.0625 − 3.59e12i)8-s + (1.71e15 + 7.09e14i)9-s + (−9.02e15 − 1.56e16i)10-s + (−5.49e16 + 3.17e16i)11-s + (5.99e16 + 1.76e17i)12-s − 4.92e17·13-s + (1.20e18 + 2.84e18i)14-s + (6.28e18 + 5.51e18i)15-s + (9.13e18 − 1.58e19i)16-s + (6.46e19 − 3.73e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.708i)2-s + (0.980 + 0.195i)3-s + (0.504 + 0.873i)4-s + (1.10 + 0.636i)5-s + (−1.06 − 0.934i)6-s + (−0.799 − 0.601i)7-s − 0.0127i·8-s + (0.923 + 0.382i)9-s + (−0.902 − 1.56i)10-s + (−1.19 + 0.690i)11-s + (0.324 + 0.955i)12-s − 0.740·13-s + (0.555 + 1.30i)14-s + (0.957 + 0.839i)15-s + (0.495 − 0.858i)16-s + (1.32 − 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.8337636770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8337636770\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.22e7 - 8.40e6i)T \) |
| 7 | \( 1 + (2.65e13 + 1.99e13i)T \) |
| good | 2 | \( 1 + (8.04e4 + 4.64e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-1.68e11 - 9.71e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (5.49e16 - 3.17e16i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 4.92e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-6.46e19 + 3.73e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (3.80e19 - 6.58e19i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (1.17e21 + 6.79e20i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 - 1.16e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (8.42e22 + 1.45e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (2.55e24 - 4.43e24i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 + 8.98e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.62e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (2.28e26 + 1.31e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (5.85e27 - 3.37e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-2.84e28 + 1.64e28i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-7.98e27 + 1.38e28i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-5.72e28 - 9.91e28i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 + 2.91e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-2.73e28 - 4.73e28i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-7.63e29 + 1.32e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 3.86e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-1.18e31 - 6.82e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 + 1.11e31T + 3.77e63T^{2} \) |
| 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
−10.48666866412275595726377386185, −10.00349681486157662990272487157, −9.463225813797134428832518230029, −7.929020649976181955089416625970, −7.04984389024479528507513242350, −5.18339166079193212780654591192, −3.23174665846082035207387714037, −2.50619747746560549853870163664, −1.66256346844357745179541705224, −0.22716634224783325646661336395,
1.01200743526263819149834368274, 2.12921779038963150246214156529, 3.28589088734220187780421976023, 5.39039453384325030290660068134, 6.43876303626964812094817682654, 7.83421855177816401606384452382, 8.575395581944296099204274759272, 9.664249662776460676869117149750, 10.05719097090541751700318005060, 12.65291958914699323510436868676
