| L(s) = 1 | + (−1.32 − 1.78i)2-s + (1.62 − 0.605i)3-s + (−0.843 + 2.81i)4-s + (2.46 − 1.61i)5-s + (−3.23 − 2.08i)6-s + (−2.02 + 2.14i)7-s + (1.96 − 0.714i)8-s + (2.26 − 1.96i)9-s + (−6.15 − 2.23i)10-s + (−4.74 + 2.38i)11-s + (0.337 + 5.07i)12-s + (−0.291 − 0.675i)13-s + (6.52 + 0.762i)14-s + (3.01 − 4.11i)15-s + (1.03 + 0.680i)16-s + (1.11 − 0.933i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 − 1.26i)2-s + (0.936 − 0.349i)3-s + (−0.421 + 1.40i)4-s + (1.10 − 0.723i)5-s + (−1.31 − 0.852i)6-s + (−0.766 + 0.812i)7-s + (0.693 − 0.252i)8-s + (0.755 − 0.655i)9-s + (−1.94 − 0.708i)10-s + (−1.42 + 0.718i)11-s + (0.0973 + 1.46i)12-s + (−0.0807 − 0.187i)13-s + (1.74 + 0.203i)14-s + (0.778 − 1.06i)15-s + (0.258 + 0.170i)16-s + (0.269 − 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.517540 - 0.645843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.517540 - 0.645843i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.62 + 0.605i)T \) |
| good | 2 | \( 1 + (1.32 + 1.78i)T + (-0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (-2.46 + 1.61i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (2.02 - 2.14i)T + (-0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (4.74 - 2.38i)T + (6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (0.291 + 0.675i)T + (-8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 0.933i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-4.21 - 3.53i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.327i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (3.53 - 0.412i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (1.08 - 0.256i)T + (27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (0.643 + 3.64i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (6.10 - 8.20i)T + (-11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.461 + 7.91i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-5.37 - 1.27i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (1.83 + 3.17i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.40 - 3.21i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (1.24 + 4.17i)T + (-50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (11.2 + 1.31i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (12.6 + 4.60i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.72 + 1.35i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.69 - 2.28i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-0.210 - 0.282i)T + (-23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (-5.34 + 1.94i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.88 - 3.87i)T + (38.4 + 89.0i)T^{2} \) |
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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
−13.52944807595768554671035285716, −12.78916658936331254711713647328, −12.14992842581348427779425997906, −10.23011626756524134155551120956, −9.633747076736671448138305135473, −8.894967124997238927297284734001, −7.69220557949311015630849301741, −5.58009878629382053279023219197, −3.02474845813800772777543713571, −1.89384681623857728317082688635,
3.01808698140405391305352749030, 5.52977386267551348546716784999, 6.87186991212357433923583583566, 7.74475484223365861659449849093, 9.041043837719162786690324697530, 9.981879741943017976484959207740, 10.54241728985093320061664607875, 13.29632504336606719228401321357, 13.79550181320775156305220707327, 14.85409137111324959438805971956
