L(s) = 1 | + (8.89 + 1.37i)3-s + (27.4 + 15.8i)5-s + (−37.6 − 65.2i)7-s + (77.2 + 24.4i)9-s + (123. − 71.1i)11-s + (96.3 − 166. i)13-s + (222. + 178. i)15-s + 325. i·17-s − 314.·19-s + (−245. − 632. i)21-s + (443. + 256. i)23-s + (188. + 326. i)25-s + (653. + 323. i)27-s + (−136. + 78.8i)29-s + (−183. + 318. i)31-s + ⋯ |
L(s) = 1 | + (0.988 + 0.152i)3-s + (1.09 + 0.633i)5-s + (−0.769 − 1.33i)7-s + (0.953 + 0.302i)9-s + (1.01 − 0.588i)11-s + (0.569 − 0.987i)13-s + (0.986 + 0.793i)15-s + 1.12i·17-s − 0.870·19-s + (−0.556 − 1.43i)21-s + (0.839 + 0.484i)23-s + (0.301 + 0.522i)25-s + (0.895 + 0.444i)27-s + (−0.162 + 0.0937i)29-s + (−0.191 + 0.331i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.02862 - 0.200692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02862 - 0.200692i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-8.89 - 1.37i)T \) |
good | 5 | \( 1 + (-27.4 - 15.8i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (37.6 + 65.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-123. + 71.1i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-96.3 + 166. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 325. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 314.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-443. - 256. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (136. - 78.8i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (183. - 318. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.73e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (342. + 197. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-360. - 624. i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.15e3 - 1.24e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 3.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.13e3 - 1.23e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.24e3 + 2.14e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.29e3 - 5.71e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (1.93e3 + 3.35e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (1.04e4 - 6.01e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.45e3 - 2.52e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.10522718411499065030381321560, −10.89973514715780212399647421785, −10.26818483103082964827789114372, −9.419434585641747906672888568709, −8.258723429004922065249860843208, −6.92214400383496279808865805691, −6.09036108134727762958885195569, −3.99045743608459420725970651985, −3.07774632852087515197614801894, −1.34914749089781402497938657969,
1.62208784451203667254856550991, 2.69851067928946843269307108473, 4.42869228188299747935743847390, 5.98985619095907696948507118061, 6.91986971240020809114678480680, 8.776799515501406155374635148116, 9.157645866389381872963051219818, 9.808125998145573287073804021500, 11.68340112968302682761216337756, 12.69599911709437751561210811043