| L(s) = 1 | + (−2.16 − 1.70i)2-s + (−0.814 − 0.580i)3-s + (1.32 + 5.46i)4-s + (−1.29 + 0.249i)5-s + (0.777 + 2.64i)6-s + (1.94 + 1.79i)7-s + (4.15 − 9.08i)8-s + (0.327 + 0.945i)9-s + (3.23 + 1.66i)10-s + (−0.534 − 0.680i)11-s + (2.08 − 5.21i)12-s + (−2.20 − 3.42i)13-s + (−1.16 − 7.20i)14-s + (1.19 + 0.547i)15-s + (−14.5 + 7.48i)16-s + (2.08 + 1.98i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 − 1.20i)2-s + (−0.470 − 0.334i)3-s + (0.662 + 2.73i)4-s + (−0.578 + 0.111i)5-s + (0.317 + 1.08i)6-s + (0.735 + 0.677i)7-s + (1.46 − 3.21i)8-s + (0.109 + 0.315i)9-s + (1.02 + 0.527i)10-s + (−0.161 − 0.205i)11-s + (0.602 − 1.50i)12-s + (−0.610 − 0.950i)13-s + (−0.311 − 1.92i)14-s + (0.309 + 0.141i)15-s + (−3.63 + 1.87i)16-s + (0.505 + 0.482i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00421151 + 0.00608320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00421151 + 0.00608320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 + 0.580i)T \) |
| 7 | \( 1 + (-1.94 - 1.79i)T \) |
| 23 | \( 1 + (4.16 + 2.37i)T \) |
| good | 2 | \( 1 + (2.16 + 1.70i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (1.29 - 0.249i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (0.534 + 0.680i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (2.20 + 3.42i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.08 - 1.98i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.858 - 0.818i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (5.72 - 1.68i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.387 + 4.05i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-6.47 + 2.24i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (4.75 - 4.12i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.05 - 0.938i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (8.33 + 4.80i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.58 - 0.361i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (2.48 - 4.81i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (7.57 + 10.6i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-3.71 - 9.26i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (1.89 - 13.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (11.7 - 2.85i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (15.4 - 0.737i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (2.53 - 2.92i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.89 + 0.753i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (6.37 + 7.35i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.38434211204641388487263337807, −10.36844975516275295470653141696, −9.718394948958401574605040886414, −8.441700007471535248754311819511, −8.003127453637025556419470002828, −7.27530731554882642830697578630, −5.73601498118818237991296740871, −4.11213633439086079770324737043, −2.81252229386533579782285113238, −1.65638977407920177257010318335,
0.007929374924326539938415934717, 1.67655951779986188626500465712, 4.35876420740114074918665556515, 5.24601755548239466307553537767, 6.36255035007591623287212804863, 7.40016251409083750712560066436, 7.76813499564993040637223501545, 8.850764583697690928604796906590, 9.744573483342341258478879601509, 10.35048387629649137022760478429
