L(s) = 1 | + 4.46·2-s + (−14.2 − 6.27i)3-s − 12.0·4-s + 59.8i·5-s + (−63.6 − 28.0i)6-s + 169. i·7-s − 196.·8-s + (164. + 179. i)9-s + 267. i·10-s + (−358. − 181. i)11-s + (172. + 75.8i)12-s − 838. i·13-s + 756. i·14-s + (375. − 854. i)15-s − 491.·16-s − 512.·17-s + ⋯ |
L(s) = 1 | + 0.788·2-s + (−0.915 − 0.402i)3-s − 0.377·4-s + 1.07i·5-s + (−0.722 − 0.317i)6-s + 1.30i·7-s − 1.08·8-s + (0.675 + 0.737i)9-s + 0.845i·10-s + (−0.892 − 0.451i)11-s + (0.345 + 0.152i)12-s − 1.37i·13-s + 1.03i·14-s + (0.431 − 0.980i)15-s − 0.479·16-s − 0.430·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.339838 + 0.719495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339838 + 0.719495i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (14.2 + 6.27i)T \) |
| 11 | \( 1 + (358. + 181. i)T \) |
good | 2 | \( 1 - 4.46T + 32T^{2} \) |
| 5 | \( 1 - 59.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 169. iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 838. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 512.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.87e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.33e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.77e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.85e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.87e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.84e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.70e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.73e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.66e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.58e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.37e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.72570375536652832797913528601, −14.96633966045740790880808038286, −13.49741333753315922154670681763, −12.55041135970528924057654196027, −11.45083319514742308814079707816, −10.10973617248327058409789935913, −8.057879940906696116716634097272, −6.11108902749414972766624663584, −5.32486068657139785671111711772, −2.95622490059246783513194538887,
0.41839507253354455158098168910, 4.34579097973186112987807189854, 4.86313399842240998522389021378, 6.73882915619198983424819411975, 8.928112025174708980656032718048, 10.29060828328713455592751562036, 11.78144721812198888853070880831, 12.93131565500976581069933502478, 13.72634347685295915527451800431, 15.31725879644629946668272436833