| L(s) = 1 | + (−0.551 + 1.63i)2-s + (−1.84 + 2.36i)3-s + (0.805 + 0.612i)4-s + (−3.83 − 1.52i)5-s + (−2.85 − 4.32i)6-s + (−0.768 + 0.355i)7-s + (−7.17 + 4.86i)8-s + (−2.19 − 8.72i)9-s + (4.61 − 5.43i)10-s + (−5.86 − 1.62i)11-s + (−2.93 + 0.775i)12-s + (−0.796 + 1.50i)13-s + (−0.158 − 1.45i)14-s + (10.6 − 6.24i)15-s + (−2.92 − 10.5i)16-s + (3.84 − 8.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.275 + 0.819i)2-s + (−0.615 + 0.788i)3-s + (0.201 + 0.153i)4-s + (−0.766 − 0.305i)5-s + (−0.476 − 0.721i)6-s + (−0.109 + 0.0507i)7-s + (−0.896 + 0.607i)8-s + (−0.243 − 0.969i)9-s + (0.461 − 0.543i)10-s + (−0.533 − 0.148i)11-s + (−0.244 + 0.0646i)12-s + (−0.0612 + 0.115i)13-s + (−0.0113 − 0.103i)14-s + (0.711 − 0.416i)15-s + (−0.182 − 0.658i)16-s + (0.226 − 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.230 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.119399 - 0.151057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119399 - 0.151057i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.84 - 2.36i)T \) |
| 59 | \( 1 + (22.7 - 54.4i)T \) |
| good | 2 | \( 1 + (0.551 - 1.63i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (3.83 + 1.52i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (0.768 - 0.355i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (5.86 + 1.62i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (0.796 - 1.50i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-3.84 + 8.31i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (11.0 - 2.44i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-18.5 - 3.04i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (5.32 + 15.8i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (18.2 + 4.02i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (20.1 - 29.7i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (63.2 - 10.3i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-9.03 - 32.5i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (45.4 - 18.1i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-28.3 + 24.1i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-50.3 - 16.9i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-13.9 - 20.5i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (63.7 - 25.3i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (117. - 12.7i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-1.48 + 0.896i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-77.9 + 4.22i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (21.6 + 64.2i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-39.8 - 4.33i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.99730172174854024821772050601, −11.87701651772832718075281739686, −11.36375804154917346597340500414, −10.13742331994602102625737107355, −8.945596114484824138706079051441, −8.045791458817634967754688839318, −6.92611756939057305277925220336, −5.81013170065134691610465201086, −4.66796827690394577895078712810, −3.20448416777182546403770283411,
0.12793679069153420656982588981, 1.89684684393170886201031874552, 3.40545989053704557580370220130, 5.30289867963252345024963437064, 6.60597085288969869737014199247, 7.44649158309492924848068789891, 8.698008572792102682415038776063, 10.23898509335059438992806421614, 10.89582611094089177442549473364, 11.66372374212636633888662297911
