| L(s) = 1 | + (−19.5 − 11.3i)2-s + (213. + 116. i)3-s + (255. + 443. i)4-s + (−1.21e3 + 698. i)5-s + (−2.86e3 − 4.69e3i)6-s + (−1.29e4 + 2.24e4i)7-s − 1.15e4i·8-s + (3.20e4 + 4.95e4i)9-s + 3.16e4·10-s + (5.36e4 + 3.09e4i)11-s + (3.14e3 + 1.24e5i)12-s + (3.79e4 + 6.57e4i)13-s + (5.07e5 − 2.92e5i)14-s + (−3.39e5 + 8.58e3i)15-s + (−1.31e5 + 2.27e5i)16-s + 8.66e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.878 + 0.477i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.368 − 0.603i)6-s + (−0.769 + 1.33i)7-s − 0.353i·8-s + (0.543 + 0.839i)9-s + 0.316·10-s + (0.333 + 0.192i)11-s + (0.0126 + 0.499i)12-s + (0.102 + 0.177i)13-s + (0.942 − 0.544i)14-s + (−0.447 + 0.0113i)15-s + (−0.125 + 0.216i)16-s + 0.610i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0684312 + 1.10283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0684312 + 1.10283i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 + 11.3i)T \) |
| 3 | \( 1 + (-213. - 116. i)T \) |
| 5 | \( 1 + (1.21e3 - 698. i)T \) |
| good | 7 | \( 1 + (1.29e4 - 2.24e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-5.36e4 - 3.09e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-3.79e4 - 6.57e4i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 8.66e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.81e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (1.05e7 - 6.08e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (1.23e7 + 7.13e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-8.75e6 - 1.51e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 1.16e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (-1.28e8 + 7.44e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-4.99e7 + 8.65e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (3.16e8 + 1.82e8i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 - 9.47e6iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-5.16e8 + 2.98e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (6.45e8 - 1.11e9i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.95e8 + 3.38e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.89e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.42e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (1.01e9 - 1.75e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (3.87e9 + 2.23e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 3.09e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-1.68e9 + 2.92e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−12.36852485638896914348467144379, −11.53263650297197909887017667220, −10.04353775715745271224999150123, −9.381463117145532671481351150596, −8.446469703404765931053285949561, −7.35720436173036723236528062695, −5.75868480282349972581115592952, −3.88506976254708466317939366558, −2.92923003609921681546558648678, −1.77689584159287655395647597611,
0.32244879331545691948283921782, 1.23728215379090708291816770223, 3.01941562797425157007896189524, 4.19080544960121815137395114786, 6.26594705421476735390362529733, 7.34512569588050227010146338486, 7.993831316217359195001377362851, 9.351103826450509348213885103191, 10.05655486098342483330024399776, 11.52538963666645152061736733248
