| L(s) = 1 | + (−2.35 + 1.21i)2-s + (−0.786 + 0.618i)3-s + (2.91 − 4.09i)4-s + (2.56 − 2.44i)5-s + (1.10 − 2.41i)6-s + (0.193 + 2.63i)7-s + (−1.14 + 7.93i)8-s + (0.235 − 0.971i)9-s + (−3.07 + 8.88i)10-s + (−4.35 − 2.24i)11-s + (0.239 + 5.01i)12-s + (2.03 + 2.34i)13-s + (−3.65 − 5.98i)14-s + (−0.505 + 3.51i)15-s + (−3.66 − 10.5i)16-s + (5.00 − 0.478i)17-s + ⋯ |
| L(s) = 1 | + (−1.66 + 0.858i)2-s + (−0.453 + 0.356i)3-s + (1.45 − 2.04i)4-s + (1.14 − 1.09i)5-s + (0.449 − 0.984i)6-s + (0.0730 + 0.997i)7-s + (−0.403 + 2.80i)8-s + (0.0785 − 0.323i)9-s + (−0.972 + 2.81i)10-s + (−1.31 − 0.676i)11-s + (0.0690 + 1.44i)12-s + (0.564 + 0.651i)13-s + (−0.978 − 1.59i)14-s + (−0.130 + 0.906i)15-s + (−0.915 − 2.64i)16-s + (1.21 − 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.649635 + 0.172802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.649635 + 0.172802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (-0.193 - 2.63i)T \) |
| 23 | \( 1 + (3.34 + 3.43i)T \) |
| good | 2 | \( 1 + (2.35 - 1.21i)T + (1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.56 + 2.44i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (4.35 + 2.24i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 2.34i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.00 + 0.478i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-3.28 - 0.313i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (0.0709 - 0.155i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-5.38 + 2.15i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-1.62 + 6.67i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (-5.88 - 1.72i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.536 - 3.73i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (1.46 - 2.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.99 - 1.92i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-2.18 + 6.30i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (1.00 + 0.792i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (0.283 - 5.96i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 2.36i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.83 + 6.79i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-15.5 + 2.99i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (2.30 - 0.676i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (8.38 + 3.35i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 2.95i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.60948917893701486291453707251, −9.795020227379527727412221242897, −9.290141540294200723893774668651, −8.476203770712797596728230513048, −7.81385744212430191299435801789, −6.21049397853348162822705537401, −5.74722124165453093204828622788, −5.08909021385261898224236775238, −2.34383713003340900933260613730, −0.940921741611715799336437036815,
1.12427677002596870198197298383, 2.37480102006399857569614834719, 3.36750638562221796832976612323, 5.53194793104233465377863492890, 6.72445609372231726104033792324, 7.54100310404662450501820667193, 8.055865764464096323346106081957, 9.653933151625517722942624159698, 10.27475066806999035992596383034, 10.42184258563460365821598639129
