| 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
−10.41499240249968776295492539358, −10.16018944895105688106889105789, −9.173549107533590911657322326701, −7.86521358846239409479147438323, −6.87845504967465973427706915164, −6.26839643818163637406346739593, −5.50634617739584942828505023023, −3.78686103580933899811205743316, −2.87939695266903988987103690656, −1.28742466996855883880092502370,
1.43203302630389266416164544170, 3.03060287061720932882668165954, 3.84016129965003103554797612644, 5.20406193174539730824894252589, 6.63965884386396092869492983189, 6.80944352131620399461762300048, 8.167742881429880580058517038567, 9.218945847780219258078443105896, 9.573811789941803201637365910601, 11.04662983564993422225812865770
