| L(s) = 1 | + (1.33 + 0.467i)2-s + (−0.123 + 1.09i)3-s + (1.56 + 1.24i)4-s + (2.16 + 4.49i)5-s + (−0.679 + 1.41i)6-s + (−6.05 − 7.58i)7-s + (1.50 + 2.39i)8-s + (7.58 + 1.73i)9-s + (0.790 + 7.01i)10-s + (7.51 − 11.9i)11-s + (−1.56 + 1.56i)12-s + (−20.2 + 4.62i)13-s + (−4.53 − 12.9i)14-s + (−5.21 + 1.82i)15-s + (0.890 + 3.89i)16-s + (−16.8 − 16.8i)17-s + ⋯ |
| L(s) = 1 | + (0.667 + 0.233i)2-s + (−0.0413 + 0.366i)3-s + (0.390 + 0.311i)4-s + (0.432 + 0.899i)5-s + (−0.113 + 0.235i)6-s + (−0.864 − 1.08i)7-s + (0.188 + 0.299i)8-s + (0.842 + 0.192i)9-s + (0.0790 + 0.701i)10-s + (0.683 − 1.08i)11-s + (−0.130 + 0.130i)12-s + (−1.55 + 0.355i)13-s + (−0.323 − 0.925i)14-s + (−0.347 + 0.121i)15-s + (0.0556 + 0.243i)16-s + (−0.993 − 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.52844 + 0.587347i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52844 + 0.587347i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 - 0.467i)T \) |
| 29 | \( 1 + (8.77 - 27.6i)T \) |
| good | 3 | \( 1 + (0.123 - 1.09i)T + (-8.77 - 2.00i)T^{2} \) |
| 5 | \( 1 + (-2.16 - 4.49i)T + (-15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (6.05 + 7.58i)T + (-10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-7.51 + 11.9i)T + (-52.4 - 109. i)T^{2} \) |
| 13 | \( 1 + (20.2 - 4.62i)T + (152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (16.8 + 16.8i)T + 289iT^{2} \) |
| 19 | \( 1 + (-11.9 + 1.34i)T + (351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (1.98 + 0.953i)T + (329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-9.99 - 3.49i)T + (751. + 599. i)T^{2} \) |
| 37 | \( 1 + (-26.7 - 42.6i)T + (-593. + 1.23e3i)T^{2} \) |
| 41 | \( 1 + (19.5 - 19.5i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-11.5 - 33.1i)T + (-1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (33.8 + 21.2i)T + (958. + 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-62.5 + 30.1i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 + 28.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-0.657 + 5.83i)T + (-3.62e3 - 828. i)T^{2} \) |
| 67 | \( 1 + (114. + 26.1i)T + (4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (36.1 - 8.24i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-38.4 + 13.4i)T + (4.16e3 - 3.32e3i)T^{2} \) |
| 79 | \( 1 + (58.2 - 36.6i)T + (2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-75.9 + 95.2i)T + (-1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (58.9 + 20.6i)T + (6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-16.9 - 150. i)T + (-9.17e3 + 2.09e3i)T^{2} \) |
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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
−14.87658746392557991439184464749, −13.92667017768302362069987165637, −13.20824818517696583923428426049, −11.64178961351546461976879982046, −10.43805799404281425780602455622, −9.526523102973328260755286254469, −7.21748119245648007155888662849, −6.57698878233114075120548411561, −4.63496062973981366607193555750, −3.11875680933496730947152752233,
2.11667350025906676025635170272, 4.45076407944202404165274957943, 5.86986668122635447896948779008, 7.19739860459422859357346282202, 9.209595186835362893603114691527, 9.953279905448899697173032149418, 12.07894742442880525859416570026, 12.52856040498472869909755735617, 13.26261625296470479820522701898, 14.91396382223230880824646792355
