| L(s) = 1 | + (0.940 − 1.05i)2-s + (−1.38 + 2.72i)3-s + (−0.229 − 1.98i)4-s + (−0.521 − 2.17i)5-s + (1.56 + 4.02i)6-s + (0.707 − 0.707i)7-s + (−2.31 − 1.62i)8-s + (−3.72 − 5.12i)9-s + (−2.78 − 1.49i)10-s + (−3.29 + 4.52i)11-s + (5.72 + 2.12i)12-s + (−0.466 + 2.94i)13-s + (−0.0814 − 1.41i)14-s + (6.64 + 1.59i)15-s + (−3.89 + 0.913i)16-s + (−0.336 + 0.171i)17-s + ⋯ |
| L(s) = 1 | + (0.665 − 0.746i)2-s + (−0.800 + 1.57i)3-s + (−0.114 − 0.993i)4-s + (−0.233 − 0.972i)5-s + (0.640 + 1.64i)6-s + (0.267 − 0.267i)7-s + (−0.818 − 0.574i)8-s + (−1.24 − 1.70i)9-s + (−0.881 − 0.472i)10-s + (−0.992 + 1.36i)11-s + (1.65 + 0.614i)12-s + (−0.129 + 0.816i)13-s + (−0.0217 − 0.377i)14-s + (1.71 + 0.411i)15-s + (−0.973 + 0.228i)16-s + (−0.0817 + 0.0416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0792863 + 0.239687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0792863 + 0.239687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.940 + 1.05i)T \) |
| 5 | \( 1 + (0.521 + 2.17i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (1.38 - 2.72i)T + (-1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (3.29 - 4.52i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.466 - 2.94i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.336 - 0.171i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.65 - 5.08i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.403 + 2.54i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (4.85 - 1.57i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.80 + 1.23i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (9.90 + 1.56i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-7.26 + 5.28i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (6.36 + 6.36i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.63 - 1.34i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-4.35 - 2.21i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.45 + 1.78i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 8.26i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.52 + 4.96i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (1.25 - 0.407i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.73 - 1.38i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.91 - 15.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.71 - 2.40i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (2.60 - 3.58i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.29 + 4.50i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.67904676079406391447464421340, −10.18254309304961783465496075445, −9.464127224623540853806268814874, −8.653293083104834903444650325794, −7.11639297824513531458061035580, −5.60242506057504972864171846401, −5.18827123226571703874782287549, −4.24647468709828200312522055357, −3.89341298450498206177232123366, −1.97126351926316539223384274807,
0.10892447123631394018583086052, 2.39095654950568373322288937464, 3.29585233256762810908054019052, 5.15622533767269610335104254531, 5.77550174734208546237964925304, 6.51811168827335091631832614229, 7.36869346078518977997906271286, 7.88070437173008213254290102287, 8.692744280382479299116263712013, 10.56569434630294132801713489612
