L(s) = 1 | + (1.29 − 0.380i)2-s + (−1.87 − 2.16i)3-s + (−0.145 + 0.0937i)4-s + (−0.142 − 0.989i)5-s + (−3.25 − 2.09i)6-s + (1.66 − 3.64i)7-s + (−1.92 + 2.21i)8-s + (−0.740 + 5.15i)9-s + (−0.561 − 1.22i)10-s + (5.32 + 1.56i)11-s + (0.476 + 0.139i)12-s + (0.162 + 0.356i)13-s + (0.771 − 5.36i)14-s + (−1.87 + 2.16i)15-s + (−1.50 + 3.29i)16-s + (−0.659 − 0.423i)17-s + ⋯ |
L(s) = 1 | + (0.916 − 0.269i)2-s + (−1.08 − 1.24i)3-s + (−0.0729 + 0.0468i)4-s + (−0.0636 − 0.442i)5-s + (−1.32 − 0.854i)6-s + (0.629 − 1.37i)7-s + (−0.680 + 0.784i)8-s + (−0.246 + 1.71i)9-s + (−0.177 − 0.388i)10-s + (1.60 + 0.471i)11-s + (0.137 + 0.0403i)12-s + (0.0451 + 0.0987i)13-s + (0.206 − 1.43i)14-s + (−0.484 + 0.558i)15-s + (−0.376 + 0.823i)16-s + (−0.159 − 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.742926 - 0.858836i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.742926 - 0.858836i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.73 + 0.783i)T \) |
good | 2 | \( 1 + (-1.29 + 0.380i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (1.87 + 2.16i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.66 + 3.64i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-5.32 - 1.56i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.162 - 0.356i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.659 + 0.423i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.661 - 0.425i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (0.130 + 0.0840i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.02 - 2.33i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.58 - 11.0i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.44 + 10.0i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.99 + 2.30i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 2.46T + 47T^{2} \) |
| 53 | \( 1 + (2.03 - 4.44i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.65 - 10.1i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (4.03 - 4.65i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (5.93 - 1.74i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.07 + 1.19i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.795 - 0.511i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.726 + 1.59i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.38 + 9.66i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.26 - 6.07i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.60 - 11.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.27905329474397858076693914572, −12.18863787905886878287707562151, −11.74494054263670280875724378622, −10.71130497760929917326412276184, −8.827076332216665504957544667590, −7.40303161100663783220346188635, −6.51454660089060569251612105258, −5.08030842561796034607063950107, −4.07051842438134741079234491632, −1.34902075979349022788815147091,
3.60138457119017634039531737455, 4.76352794637077942523390041845, 5.70011820969388663597845854762, 6.47436954094112311860405956373, 8.895096401970780793858384305919, 9.602962061165700151556167176861, 11.12135258419788296464268277340, 11.63860525467620918339082520805, 12.65822891924038527271888277569, 14.28999128629930698226986529465