| 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
−15.86829841619820142932129139221, −14.50881665496280593939047463716, −13.95606200877136385627120203020, −12.81987671275366949996742481574, −10.80447113098320663360557727118, −9.960606142627451961606024181512, −7.58872807885372825269803528739, −6.32179423267374692292296099006, −5.42020692487578127624069153973, −2.19958744669068319328347381727,
2.12893039124659693018632945250, 4.41747696378330501708465725834, 5.74445589454044463695839053079, 8.245814209951503255196805334315, 9.775998025013691289381241102717, 10.84222650785460233174221219923, 12.45958511586584059506007872785, 13.15933588609348219209911987491, 14.33832644187693850983561531447, 16.17254205952249831182211873840
