| L(s) = 1 | + (−1.87 − 0.550i)2-s + (0.142 − 0.989i)3-s + (1.52 + 0.981i)4-s + (−1.74 − 3.82i)5-s + (−0.811 + 1.77i)6-s + (−0.0581 − 0.0170i)7-s + (0.235 + 0.272i)8-s + (−0.959 − 0.281i)9-s + (1.16 + 8.11i)10-s + (−0.281 − 0.615i)11-s + (1.18 − 1.37i)12-s + (−1.51 + 1.74i)13-s + (0.0995 + 0.0639i)14-s + (−4.02 + 1.18i)15-s + (−1.80 − 3.94i)16-s + (−1.61 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 − 0.389i)2-s + (0.0821 − 0.571i)3-s + (0.763 + 0.490i)4-s + (−0.780 − 1.70i)5-s + (−0.331 + 0.725i)6-s + (−0.0219 − 0.00644i)7-s + (0.0833 + 0.0962i)8-s + (−0.319 − 0.0939i)9-s + (0.369 + 2.56i)10-s + (−0.0847 − 0.185i)11-s + (0.343 − 0.396i)12-s + (−0.420 + 0.485i)13-s + (0.0265 + 0.0170i)14-s + (−1.04 + 0.305i)15-s + (−0.450 − 0.985i)16-s + (−0.391 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0536791 + 0.319990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0536791 + 0.319990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (7.54 + 3.17i)T \) |
| good | 2 | \( 1 + (1.87 + 0.550i)T + (1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (1.74 + 3.82i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (0.0581 + 0.0170i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (0.281 + 0.615i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (1.51 - 1.74i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.61 - 1.03i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.944 + 0.277i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (0.204 - 1.42i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 + (5.96 + 6.88i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + (-6.75 + 4.34i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.95 + 4.47i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.442 + 3.08i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (1.29 + 0.833i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.974 - 1.12i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-3.67 + 8.05i)T + (-39.9 - 46.1i)T^{2} \) |
| 71 | \( 1 + (11.1 + 7.15i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.58 - 3.48i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.64 + 4.20i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.05 - 4.49i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.724 - 5.04i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + 1.45T + 97T^{2} \) |
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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
−11.71268743635413501144048112012, −11.09269190477140894949885189849, −9.399528306122511405111675077876, −9.101782423678334573941832301878, −7.980575791566417288502026397019, −7.47741645296081441862824144520, −5.53955958436237598613034399442, −4.18745757907941614196100783564, −1.89219170656595194662900480883, −0.42232341719512610795802060857,
2.79614337803184667232073083806, 4.16282414933951150201701302830, 6.16014224625147232579086896413, 7.36034978744734867921025232668, 7.75911979003335404065520460044, 9.134282338228024059099545818957, 10.00414294121875754522058900355, 10.83547850464920203270222866598, 11.36370166223052413038452857910, 12.93156451107770146429014422561
