| L(s) = 1 | + (0.0265 + 0.252i)2-s + (1.89 − 0.402i)4-s + (−2.90 − 0.305i)5-s + (−2.02 + 1.81i)7-s + (0.308 + 0.949i)8-s − 0.741i·10-s + (−0.283 − 3.30i)11-s + (−1.16 + 2.60i)13-s + (−0.512 − 0.461i)14-s + (3.30 − 1.47i)16-s + (−3.60 − 2.62i)17-s + (−1.81 + 0.590i)19-s + (−5.62 + 0.591i)20-s + (0.826 − 0.159i)22-s + (−0.706 + 0.408i)23-s + ⋯ |
| L(s) = 1 | + (0.0187 + 0.178i)2-s + (0.946 − 0.201i)4-s + (−1.29 − 0.136i)5-s + (−0.763 + 0.687i)7-s + (0.109 + 0.335i)8-s − 0.234i·10-s + (−0.0855 − 0.996i)11-s + (−0.322 + 0.723i)13-s + (−0.136 − 0.123i)14-s + (0.826 − 0.367i)16-s + (−0.875 − 0.636i)17-s + (−0.416 + 0.135i)19-s + (−1.25 + 0.132i)20-s + (0.176 − 0.0339i)22-s + (−0.147 + 0.0851i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0331783 - 0.163881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0331783 - 0.163881i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.283 + 3.30i)T \) |
| good | 2 | \( 1 + (-0.0265 - 0.252i)T + (-1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (2.90 + 0.305i)T + (4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (2.02 - 1.81i)T + (0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (1.16 - 2.60i)T + (-8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (3.60 + 2.62i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.81 - 0.590i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.706 - 0.408i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.38 + 7.09i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (5.46 + 2.43i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.83 + 5.65i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (5.64 - 6.26i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (10.2 + 5.93i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.60 - 7.54i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-6.14 - 8.45i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0193 - 0.0912i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.872 - 1.95i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.703 - 1.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.12 + 2.91i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.67 + 0.543i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.09 + 0.745i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (7.15 - 3.18i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 - 2.06iT - 89T^{2} \) |
| 97 | \( 1 + (0.174 + 1.66i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−9.665251754678629272576230911132, −8.850802940611434154154195092681, −7.951334403320449493185224455654, −7.19878068122660948095996048809, −6.35029173788423981179663878753, −5.57372839088805263855883441078, −4.23967318329062012108009560353, −3.24542906869019432512700055395, −2.17797946375371219446878501334, −0.07103472197065289679440718213,
1.94231134422450120207643283452, 3.35324419572364905580932502261, 3.84653388447162257336330205989, 5.10460218435627006266360072768, 6.64682673644879060205919963047, 7.02787674122658999804636969194, 7.76592905226722039888616211683, 8.642163414271731820620777847834, 9.999736882435085361890750534621, 10.51330664759493279875647394950
