| 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
−12.69599911709437751561210811043, −11.68340112968302682761216337756, −9.808125998145573287073804021500, −9.157645866389381872963051219818, −8.776799515501406155374635148116, −6.91986971240020809114678480680, −5.98985619095907696948507118061, −4.42869228188299747935743847390, −2.69851067928946843269307108473, −1.62208784451203667254856550991,
1.34914749089781402497938657969, 3.07774632852087515197614801894, 3.99045743608459420725970651985, 6.09036108134727762958885195569, 6.92214400383496279808865805691, 8.258723429004922065249860843208, 9.419434585641747906672888568709, 10.26818483103082964827789114372, 10.89973514715780212399647421785, 13.10522718411499065030381321560
