L(s) = 1 | + (2.24 − 1.62i)2-s + (−0.309 − 0.951i)3-s + (1.75 − 5.40i)4-s + (−2.24 − 1.62i)6-s + (0.703 − 2.16i)7-s + (−3.15 − 9.71i)8-s + (−0.809 + 0.587i)9-s + (−0.105 + 3.31i)11-s − 5.68·12-s + (−0.352 + 0.256i)13-s + (−1.95 − 6.00i)14-s + (−13.7 − 9.96i)16-s + (4.04 + 2.93i)17-s + (−0.856 + 2.63i)18-s + (1.45 + 4.46i)19-s + ⋯ |
L(s) = 1 | + (1.58 − 1.15i)2-s + (−0.178 − 0.549i)3-s + (0.878 − 2.70i)4-s + (−0.915 − 0.665i)6-s + (0.266 − 0.818i)7-s + (−1.11 − 3.43i)8-s + (−0.269 + 0.195i)9-s + (−0.0317 + 0.999i)11-s − 1.64·12-s + (−0.0977 + 0.0710i)13-s + (−0.521 − 1.60i)14-s + (−3.42 − 2.49i)16-s + (0.981 + 0.712i)17-s + (−0.201 + 0.621i)18-s + (0.332 + 1.02i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.454837 - 3.49724i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.454837 - 3.49724i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (0.105 - 3.31i)T \) |
good | 2 | \( 1 + (-2.24 + 1.62i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.703 + 2.16i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.352 - 0.256i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.04 - 2.93i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.45 - 4.46i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 0.845T + 23T^{2} \) |
| 29 | \( 1 + (0.821 - 2.52i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.77 + 2.74i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.73 + 8.42i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.32 + 4.08i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 + (0.144 + 0.445i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.76 - 6.37i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.21 + 3.74i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.39 + 1.74i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + (-9.15 - 6.65i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.60 - 8.01i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.79 + 6.38i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.78 + 3.47i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + (6.99 - 5.08i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.24964141404726256947935600816, −9.535987226311662847464090498601, −7.81970572335325383446375763644, −6.96829543205724313296944129423, −6.01275652638458296395046560158, −5.18954517220457016304041743955, −4.24219274713163718573979944734, −3.43813190745573116827641858160, −2.10220126901709932379634924783, −1.18616893066572821877535260385,
2.78628428175166812179378958449, 3.39263954382239583992676463162, 4.75437235309880314084245417525, 5.20021916392517406043983060158, 6.03921183588100069740940177808, 6.81003771687662692132913602305, 7.961670127673698034376231375547, 8.521954772274143716006061132572, 9.581772447163898172125197710297, 11.07486922485139381097811416500