L(s) = 1 | + (2.51 + 4.35i)2-s + (4.79 − 1.99i)3-s + (−8.61 + 14.9i)4-s + (20.7 + 15.8i)6-s + (2.69 + 4.66i)7-s − 46.4·8-s + (19.0 − 19.1i)9-s + (19.5 + 33.8i)11-s + (−11.5 + 88.8i)12-s + (−43.3 + 75.0i)13-s + (−13.5 + 23.4i)14-s + (−47.6 − 82.4i)16-s + 15.4·17-s + (131. + 34.6i)18-s − 26.8·19-s + ⋯ |
L(s) = 1 | + (0.888 + 1.53i)2-s + (0.923 − 0.384i)3-s + (−1.07 + 1.86i)4-s + (1.41 + 1.07i)6-s + (0.145 + 0.251i)7-s − 2.05·8-s + (0.704 − 0.709i)9-s + (0.535 + 0.928i)11-s + (−0.277 + 2.13i)12-s + (−0.924 + 1.60i)13-s + (−0.258 + 0.446i)14-s + (−0.744 − 1.28i)16-s + 0.219·17-s + (1.71 + 0.453i)18-s − 0.324·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.06226 + 3.38875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06226 + 3.38875i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.79 + 1.99i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.51 - 4.35i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-2.69 - 4.66i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-19.5 - 33.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (43.3 - 75.0i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-55.5 + 96.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (24.6 + 42.6i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-89.9 + 155. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (13.8 - 23.9i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (30.2 + 52.3i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (48.1 + 83.3i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 251.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-38.4 + 66.5i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (245. + 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-119. + 206. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 640.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 769.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (331. + 573. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-651. - 1.12e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 995.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-407. - 705. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.53671679174302983807868832912, −11.82686271716126590774702778443, −9.699356424341200121181020986293, −8.912186250491321554619544355330, −7.87718321179021042105021916512, −7.01496166952660334154056919959, −6.38517110837820761476782744877, −4.74591375358799176192040287307, −4.02379117558964563781633386064, −2.27179990763829296685727652133,
1.09778990511150579956733406734, 2.71545742517123333872573657817, 3.42979051421083818282430930650, 4.58820055170012245853434631034, 5.63736885795460478260147930956, 7.59038228271174231086301861662, 8.806740280073500271622841032381, 9.848811756393386309585171004518, 10.51728164126824982430962057911, 11.37192322245696830144541087424