| L(s) = 1 | + (−0.439 + 2.49i)2-s + (−1.53 − 1.28i)3-s + (−4.14 − 1.50i)4-s + (3.64 − 1.32i)5-s + (3.87 − 3.25i)6-s + (1.76 + 3.05i)7-s + (3.05 − 5.28i)8-s + (0.173 + 0.984i)9-s + (1.70 + 9.67i)10-s + (−0.5 + 0.866i)11-s + (4.41 + 7.64i)12-s + (2.52 − 2.11i)13-s + (−8.40 + 3.05i)14-s + (−7.29 − 2.65i)15-s + (5.08 + 4.26i)16-s + (−0.602 + 3.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.310 + 1.76i)2-s + (−0.884 − 0.742i)3-s + (−2.07 − 0.754i)4-s + (1.63 − 0.593i)5-s + (1.58 − 1.32i)6-s + (0.667 + 1.15i)7-s + (1.07 − 1.86i)8-s + (0.0578 + 0.328i)9-s + (0.539 + 3.05i)10-s + (−0.150 + 0.261i)11-s + (1.27 + 2.20i)12-s + (0.699 − 0.586i)13-s + (−2.24 + 0.817i)14-s + (−1.88 − 0.685i)15-s + (1.27 + 1.06i)16-s + (−0.146 + 0.828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.690158 + 0.674351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.690158 + 0.674351i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-2.77 - 3.35i)T \) |
| good | 2 | \( 1 + (0.439 - 2.49i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (1.53 + 1.28i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.64 + 1.32i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.76 - 3.05i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-2.52 + 2.11i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.602 - 3.41i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-4.91 - 1.78i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.162 + 0.921i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.65T + 37T^{2} \) |
| 41 | \( 1 + (2.61 + 2.19i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.992 + 0.361i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.13 + 6.45i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.77 - 1.73i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.01 - 11.4i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.09 + 0.400i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.52 + 14.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (14.2 - 5.18i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (10.5 + 8.81i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.38 + 6.19i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.51 + 4.35i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.922 - 0.774i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.50 + 8.54i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.99762430667419138557279017924, −11.92497616738799878226776895496, −10.37336969628416152029321884551, −9.174610512363615857399684470589, −8.593369462795595458901037201668, −7.37613680330415427295481449413, −6.04614368989806019716807755244, −5.80003586658514728371235040567, −5.08947409670719207392649138060, −1.53284915080034995178520983167,
1.35501342584562328238911062194, 2.91798988692377714922956049372, 4.45523798224992349543546290800, 5.36052644340071446066576825767, 6.93295005909695671024909872556, 8.865187050243357362181964695981, 9.747110273422990400078154833150, 10.46592509540543772361999836284, 11.01424480443162884684197258111, 11.49615732255113055214078766537
