| L(s) = 1 | + (−19.5 − 11.3i)2-s + (239. + 41.0i)3-s + (255. + 443. i)4-s + (1.21e3 − 698. i)5-s + (−4.22e3 − 3.51e3i)6-s + (−5.11e3 + 8.85e3i)7-s − 1.15e4i·8-s + (5.56e4 + 1.96e4i)9-s − 3.16e4·10-s + (−8.38e4 − 4.84e4i)11-s + (4.30e4 + 1.16e5i)12-s + (−5.74e4 − 9.95e4i)13-s + (2.00e5 − 1.15e5i)14-s + (3.18e5 − 1.17e5i)15-s + (−1.31e5 + 2.27e5i)16-s + 1.26e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.985 + 0.169i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.543 − 0.452i)6-s + (−0.304 + 0.526i)7-s − 0.353i·8-s + (0.942 + 0.333i)9-s − 0.316·10-s + (−0.520 − 0.300i)11-s + (0.173 + 0.469i)12-s + (−0.154 − 0.268i)13-s + (0.372 − 0.215i)14-s + (0.419 − 0.154i)15-s + (−0.125 + 0.216i)16-s + 0.893i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.23552 + 0.610063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23552 + 0.610063i\) |
| \(L(6)\) |
|
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 | \( 1 + (-239. - 41.0i)T \) |
| 5 | \( 1 + (-1.21e3 + 698. i)T \) |
| good | 7 | \( 1 + (5.11e3 - 8.85e3i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (8.38e4 + 4.84e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (5.74e4 + 9.95e4i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 1.26e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 1.73e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-6.63e6 + 3.83e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-2.45e7 - 1.41e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-1.05e6 - 1.81e6i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 1.00e8T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.09e8 - 6.29e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (4.70e7 - 8.14e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-1.27e8 - 7.37e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 5.19e6iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-3.10e8 + 1.79e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-3.95e8 + 6.85e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-5.84e8 - 1.01e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 - 2.00e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.30e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (1.40e8 - 2.42e8i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-3.27e9 - 1.88e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 6.97e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-5.07e9 + 8.79e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−12.33592717623556019481833188501, −10.72665187418906389522966071415, −9.881977504408783087192951664342, −8.849866707138107118657878149451, −8.156433345575530959015561389498, −6.75959783467632599858136598812, −5.08217211701456643598716470690, −3.37167767017901021569518624884, −2.44097983746974506197679898100, −1.12880530149494845750014073389,
0.71718843525278712890852349722, 2.10494136892572125953902891586, 3.31637919779456287646410844574, 5.05248754297130104722446274752, 6.81432326437866300901655711754, 7.43579882356989441040952971449, 8.713484334015986120436924214663, 9.690781446585444505615985007913, 10.43185493527937240734466825589, 12.00375113604477052475651374089
