| L(s) = 1 | + (−2.00 + 0.436i)3-s + (−2.22 − 0.223i)5-s + (1.76 − 0.964i)7-s + (1.10 − 0.504i)9-s + (−2.16 − 1.87i)11-s + (−0.486 + 0.890i)13-s + (4.56 − 0.521i)15-s + (−2.26 − 1.69i)17-s + (−0.358 + 2.49i)19-s + (−3.12 + 2.70i)21-s + (4.78 − 0.377i)23-s + (4.89 + 0.996i)25-s + (2.93 − 2.19i)27-s + (−4.90 + 0.704i)29-s + (5.20 − 3.34i)31-s + ⋯ |
| L(s) = 1 | + (−1.15 + 0.251i)3-s + (−0.994 − 0.100i)5-s + (0.667 − 0.364i)7-s + (0.368 − 0.168i)9-s + (−0.651 − 0.564i)11-s + (−0.134 + 0.247i)13-s + (1.17 − 0.134i)15-s + (−0.549 − 0.411i)17-s + (−0.0823 + 0.572i)19-s + (−0.681 + 0.590i)21-s + (0.996 − 0.0786i)23-s + (0.979 + 0.199i)25-s + (0.564 − 0.422i)27-s + (−0.910 + 0.130i)29-s + (0.934 − 0.600i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.589254 + 0.289742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589254 + 0.289742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.22 + 0.223i)T \) |
| 23 | \( 1 + (-4.78 + 0.377i)T \) |
| good | 3 | \( 1 + (2.00 - 0.436i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-1.76 + 0.964i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (2.16 + 1.87i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.486 - 0.890i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.26 + 1.69i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.358 - 2.49i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.90 - 0.704i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.20 + 3.34i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.250 + 0.0934i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.31 - 7.26i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.41 - 11.1i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-5.27 - 5.27i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.81 - 10.6i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.07 + 3.66i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.34 - 2.08i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-12.7 - 0.910i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-0.925 - 1.06i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-5.68 - 7.59i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-2.74 + 0.806i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.30 + 8.87i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.88 - 1.21i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.68 + 7.19i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.58508983682708406901300192101, −9.448574918776095130149390191897, −8.331182632197153614463837757049, −7.72981173284673936090142985625, −6.75298048746582633214891876874, −5.75389065737451519779272687779, −4.83276506829852760477847436442, −4.27027546042182488210684932454, −2.87630510238946604816974380715, −0.898473448907594926191490280771,
0.50566414522149499564327462935, 2.27059376200990576963038799156, 3.73815784194310396903748807035, 4.99205659171859398321872621310, 5.31790840198020990210401445981, 6.69581305193848861833133974105, 7.22215472459458695069351671791, 8.249300012456474999496607081301, 8.938994169280938569739860421794, 10.35662373451204844992190511975
