| L(s) = 1 | + (1.95 − 2.68i)2-s + (−1.60 + 4.94i)3-s + (1.52 + 4.70i)4-s + (35.4 − 25.7i)5-s + (10.1 + 13.9i)6-s + (8.04 − 2.61i)7-s + (66.2 + 21.5i)8-s + (−21.8 − 15.8i)9-s − 145. i·10-s + (−120. − 4.39i)11-s − 25.6·12-s + (−37.8 + 52.0i)13-s + (8.69 − 26.7i)14-s + (70.3 + 216. i)15-s + (123. − 89.5i)16-s + (165. + 227. i)17-s + ⋯ |
| L(s) = 1 | + (0.488 − 0.672i)2-s + (−0.178 + 0.549i)3-s + (0.0955 + 0.293i)4-s + (1.41 − 1.02i)5-s + (0.282 + 0.388i)6-s + (0.164 − 0.0533i)7-s + (1.03 + 0.336i)8-s + (−0.269 − 0.195i)9-s − 1.45i·10-s + (−0.999 − 0.0363i)11-s − 0.178·12-s + (−0.223 + 0.307i)13-s + (0.0443 − 0.136i)14-s + (0.312 + 0.962i)15-s + (0.481 − 0.349i)16-s + (0.571 + 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.99933 - 0.426199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99933 - 0.426199i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 - 4.94i)T \) |
| 11 | \( 1 + (120. + 4.39i)T \) |
| good | 2 | \( 1 + (-1.95 + 2.68i)T + (-4.94 - 15.2i)T^{2} \) |
| 5 | \( 1 + (-35.4 + 25.7i)T + (193. - 594. i)T^{2} \) |
| 7 | \( 1 + (-8.04 + 2.61i)T + (1.94e3 - 1.41e3i)T^{2} \) |
| 13 | \( 1 + (37.8 - 52.0i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-165. - 227. i)T + (-2.58e4 + 7.94e4i)T^{2} \) |
| 19 | \( 1 + (412. + 134. i)T + (1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + 761.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (42.8 - 13.9i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 31 | \( 1 + (454. + 330. i)T + (2.85e5 + 8.78e5i)T^{2} \) |
| 37 | \( 1 + (287. + 884. i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 41 | \( 1 + (-3.01e3 - 978. i)T + (2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 - 2.09e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.04e3 + 3.20e3i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (1.04e3 + 758. i)T + (2.43e6 + 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-879. - 2.70e3i)T + (-9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + (167. + 230. i)T + (-4.27e6 + 1.31e7i)T^{2} \) |
| 67 | \( 1 - 5.39e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-4.23e3 + 3.07e3i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (3.94e3 - 1.28e3i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (2.50e3 - 3.44e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-69.8 - 96.0i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 - 3.72e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (7.00e3 + 5.09e3i)T + (2.73e7 + 8.41e7i)T^{2} \) |
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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
−16.17254205952249831182211873840, −14.33832644187693850983561531447, −13.15933588609348219209911987491, −12.45958511586584059506007872785, −10.84222650785460233174221219923, −9.775998025013691289381241102717, −8.245814209951503255196805334315, −5.74445589454044463695839053079, −4.41747696378330501708465725834, −2.12893039124659693018632945250,
2.19958744669068319328347381727, 5.42020692487578127624069153973, 6.32179423267374692292296099006, 7.58872807885372825269803528739, 9.960606142627451961606024181512, 10.80447113098320663360557727118, 12.81987671275366949996742481574, 13.95606200877136385627120203020, 14.50881665496280593939047463716, 15.86829841619820142932129139221
