L(s) = 1 | + (1.61 − 1.61i)2-s + (−1.37 + 1.05i)3-s − 3.19i·4-s + (−0.511 − 1.90i)5-s + (−0.506 + 3.91i)6-s + (−1.48 − 2.19i)7-s + (−1.92 − 1.92i)8-s + (0.764 − 2.90i)9-s + (−3.90 − 2.25i)10-s + (1.51 − 0.404i)11-s + (3.38 + 4.38i)12-s + (−3.19 − 1.67i)13-s + (−5.92 − 1.14i)14-s + (2.71 + 2.07i)15-s + 0.173·16-s + 4.97·17-s + ⋯ |
L(s) = 1 | + (1.13 − 1.13i)2-s + (−0.792 + 0.610i)3-s − 1.59i·4-s + (−0.228 − 0.853i)5-s + (−0.206 + 1.59i)6-s + (−0.559 − 0.828i)7-s + (−0.682 − 0.682i)8-s + (0.254 − 0.967i)9-s + (−1.23 − 0.712i)10-s + (0.455 − 0.122i)11-s + (0.975 + 1.26i)12-s + (−0.884 − 0.465i)13-s + (−1.58 − 0.307i)14-s + (0.702 + 0.536i)15-s + 0.0434·16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.667055 - 1.43963i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.667055 - 1.43963i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.37 - 1.05i)T \) |
| 7 | \( 1 + (1.48 + 2.19i)T \) |
| 13 | \( 1 + (3.19 + 1.67i)T \) |
good | 2 | \( 1 + (-1.61 + 1.61i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.511 + 1.90i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.51 + 0.404i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 + (0.803 - 2.99i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 0.470T + 23T^{2} \) |
| 29 | \( 1 + (3.57 - 2.06i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.68 - 6.28i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.21 + 1.21i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.86 - 0.767i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.22 - 3.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-11.3 + 3.04i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.73 + 3.31i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.16 + 5.16i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.50 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (15.1 - 4.07i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 4.94i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (3.37 + 0.904i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.83 - 4.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.39 + 7.39i)T + 89iT^{2} \) |
| 97 | \( 1 + (-10.3 + 2.78i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.83046729529394586506339798311, −10.67957319390527955854638885167, −10.17839690342584031790243241328, −9.190121008691176149259620341856, −7.47828311541927808522292870971, −5.95831006967552392768353267778, −5.06676959926116452347541367754, −4.16922333776625917085480904307, −3.30446453932887491782196649325, −1.02024720192773573938503038813,
2.71790014719886745903394388184, 4.25255159338576754446482442967, 5.53598429770508466851570947062, 6.15111620893374675058012552049, 7.15048809103728269819147042573, 7.58747337729276357515549200597, 9.273573693922278227962714365000, 10.60243390992686763180459212321, 11.84494980519861754064502919638, 12.29110846471368351188623537696