| L(s) = 1 | + (−0.257 − 1.21i)3-s + (−1.69 + 2.93i)5-s + (9.33 + 4.15i)7-s + (6.82 − 3.03i)9-s + (3.52 − 0.370i)11-s + (11.8 − 10.6i)13-s + (3.98 + 1.29i)15-s + (−7.31 − 0.769i)17-s + (−1.19 + 1.32i)19-s + (2.62 − 12.3i)21-s + (−20.0 + 27.6i)23-s + (6.77 + 11.7i)25-s + (−11.9 − 16.4i)27-s + (29.1 − 9.48i)29-s + (−21.4 + 22.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.0857 − 0.403i)3-s + (−0.338 + 0.586i)5-s + (1.33 + 0.593i)7-s + (0.758 − 0.337i)9-s + (0.320 − 0.0337i)11-s + (0.910 − 0.820i)13-s + (0.265 + 0.0862i)15-s + (−0.430 − 0.0452i)17-s + (−0.0626 + 0.0695i)19-s + (0.125 − 0.588i)21-s + (−0.873 + 1.20i)23-s + (0.270 + 0.469i)25-s + (−0.443 − 0.610i)27-s + (1.00 − 0.327i)29-s + (−0.693 + 0.720i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0163i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.52651 + 0.0124582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52651 + 0.0124582i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (21.4 - 22.3i)T \) |
| good | 3 | \( 1 + (0.257 + 1.21i)T + (-8.22 + 3.66i)T^{2} \) |
| 5 | \( 1 + (1.69 - 2.93i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-9.33 - 4.15i)T + (32.7 + 36.4i)T^{2} \) |
| 11 | \( 1 + (-3.52 + 0.370i)T + (118. - 25.1i)T^{2} \) |
| 13 | \( 1 + (-11.8 + 10.6i)T + (17.6 - 168. i)T^{2} \) |
| 17 | \( 1 + (7.31 + 0.769i)T + (282. + 60.0i)T^{2} \) |
| 19 | \( 1 + (1.19 - 1.32i)T + (-37.7 - 359. i)T^{2} \) |
| 23 | \( 1 + (20.0 - 27.6i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (-29.1 + 9.48i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 + (-20.0 + 11.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (71.7 + 15.2i)T + (1.53e3 + 683. i)T^{2} \) |
| 43 | \( 1 + (47.0 + 42.3i)T + (193. + 1.83e3i)T^{2} \) |
| 47 | \( 1 + (-3.20 + 9.86i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (16.3 + 36.7i)T + (-1.87e3 + 2.08e3i)T^{2} \) |
| 59 | \( 1 + (-9.27 + 1.97i)T + (3.18e3 - 1.41e3i)T^{2} \) |
| 61 | \( 1 + 73.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (29.7 - 51.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (18.1 - 8.09i)T + (3.37e3 - 3.74e3i)T^{2} \) |
| 73 | \( 1 + (-33.4 + 3.51i)T + (5.21e3 - 1.10e3i)T^{2} \) |
| 79 | \( 1 + (14.7 + 1.54i)T + (6.10e3 + 1.29e3i)T^{2} \) |
| 83 | \( 1 + (21.4 - 100. i)T + (-6.29e3 - 2.80e3i)T^{2} \) |
| 89 | \( 1 + (32.9 + 45.3i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-81.3 + 59.1i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−13.16539045509779471226245729444, −11.96938111083260464531391275746, −11.30262922071473095420034500815, −10.21150302333549152628280814599, −8.694850813139872145210453733636, −7.77284675356588068515649299321, −6.63848589488225247567474643920, −5.26868616810817013963127226988, −3.66391680083640461070608323283, −1.65144339304195902386274157094,
1.51638688504187237440929604820, 4.21944348782606370974103446086, 4.71003533315321635179810074882, 6.57809931612287904928038869624, 7.958019683439159409412889693364, 8.765561537309526823414062752061, 10.19452265184352668215979291867, 11.09664920849481537172320829638, 11.98340385735471439952731883740, 13.25197536671049157879864739469
