| L(s) = 1 | + (0.0448 − 0.123i)2-s + (1.51 + 1.27i)4-s + (0.231 − 1.31i)5-s + (−2.62 + 0.321i)7-s + (0.452 − 0.261i)8-s + (−0.151 − 0.0873i)10-s + (4.31 − 0.759i)11-s + (1.38 + 3.81i)13-s + (−0.0781 + 0.337i)14-s + (0.676 + 3.83i)16-s + (1.85 − 3.21i)17-s + (4.31 − 2.49i)19-s + (2.02 − 1.69i)20-s + (0.0996 − 0.565i)22-s + (−0.242 + 0.289i)23-s + ⋯ |
| L(s) = 1 | + (0.0317 − 0.0871i)2-s + (0.759 + 0.637i)4-s + (0.103 − 0.586i)5-s + (−0.992 + 0.121i)7-s + (0.159 − 0.0923i)8-s + (−0.0478 − 0.0276i)10-s + (1.29 − 0.229i)11-s + (0.385 + 1.05i)13-s + (−0.0208 + 0.0903i)14-s + (0.169 + 0.959i)16-s + (0.450 − 0.780i)17-s + (0.990 − 0.571i)19-s + (0.452 − 0.379i)20-s + (0.0212 − 0.120i)22-s + (−0.0505 + 0.0603i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79114 + 0.156096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79114 + 0.156096i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.321i)T \) |
| good | 2 | \( 1 + (-0.0448 + 0.123i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.231 + 1.31i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-4.31 + 0.759i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.38 - 3.81i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 3.21i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.31 + 2.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.242 - 0.289i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.57 - 4.31i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.693 + 0.826i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.172 - 0.298i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 - 1.85i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.390 + 2.21i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (9.77 - 8.19i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 9.19iT - 53T^{2} \) |
| 59 | \( 1 + (-2.24 + 12.7i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.14 + 1.36i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 + 1.96i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.30 + 1.33i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.63 - 2.67i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.48 - 1.99i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 1.46i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.5 - 2.55i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.04662983564993422225812865770, −9.573811789941803201637365910601, −9.218945847780219258078443105896, −8.167742881429880580058517038567, −6.80944352131620399461762300048, −6.63965884386396092869492983189, −5.20406193174539730824894252589, −3.84016129965003103554797612644, −3.03060287061720932882668165954, −1.43203302630389266416164544170,
1.28742466996855883880092502370, 2.87939695266903988987103690656, 3.78686103580933899811205743316, 5.50634617739584942828505023023, 6.26839643818163637406346739593, 6.87845504967465973427706915164, 7.86521358846239409479147438323, 9.173549107533590911657322326701, 10.16018944895105688106889105789, 10.41499240249968776295492539358
