| L(s) = 1 | + (0.685 − 1.58i)2-s + (−0.684 − 0.725i)4-s + (−0.0800 − 1.37i)5-s + (3.19 + 0.758i)7-s + (1.63 − 0.593i)8-s + (−2.24 − 0.815i)10-s + (−0.171 − 0.112i)11-s + (−5.21 − 0.610i)13-s + (3.39 − 4.56i)14-s + (0.290 − 4.99i)16-s + (−3.88 + 3.26i)17-s + (−2.25 − 1.88i)19-s + (−0.943 + 0.999i)20-s + (−0.297 + 0.195i)22-s + (3.66 − 0.867i)23-s + ⋯ |
| L(s) = 1 | + (0.484 − 1.12i)2-s + (−0.342 − 0.362i)4-s + (−0.0358 − 0.614i)5-s + (1.20 + 0.286i)7-s + (0.576 − 0.209i)8-s + (−0.708 − 0.257i)10-s + (−0.0517 − 0.0340i)11-s + (−1.44 − 0.169i)13-s + (0.908 − 1.22i)14-s + (0.0726 − 1.24i)16-s + (−0.942 + 0.791i)17-s + (−0.516 − 0.433i)19-s + (−0.210 + 0.223i)20-s + (−0.0633 + 0.0416i)22-s + (0.763 − 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0347 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0347 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21376 - 1.25672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21376 - 1.25672i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.685 + 1.58i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.0800 + 1.37i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-3.19 - 0.758i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (0.171 + 0.112i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (5.21 + 0.610i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (3.88 - 3.26i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.25 + 1.88i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.66 + 0.867i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-3.76 - 5.05i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (1.45 - 4.86i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 8.19i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-1.09 - 2.53i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (1.41 - 0.712i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (0.596 + 1.99i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (4.51 + 7.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (12.4 - 8.15i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.03 + 1.09i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (8.00 - 10.7i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (4.39 + 1.59i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.3 + 5.56i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (0.588 - 1.36i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-5.64 + 13.0i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-4.83 + 1.75i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.104 + 1.79i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−11.96145508935366746512364764445, −11.03759848959615777324990431430, −10.35680506049896910160986984977, −9.014765526877439925193476627170, −8.115735327457488551121079189845, −6.88101763891125525714947557190, −4.89207024910907088358452510619, −4.67536798302095278512463035844, −2.85715474000987708478908707530, −1.60176484700182108503494517504,
2.28774015336641210072786683004, 4.40349448887401982796378003683, 5.08332758938584398806264200919, 6.41743842441336242459049828382, 7.34320951926805415132560151689, 7.910179315300792640309374090710, 9.304621381995164127376380017394, 10.68133663000384302155699590188, 11.24464465840072419813414735946, 12.48779163443228500026295603513
