L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 0.999·6-s + (−1.06 + 0.775i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.567 + 0.412i)11-s + (0.309 + 0.951i)12-s + (0.501 − 1.54i)13-s + (1.06 + 0.775i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−4.24 − 3.08i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.178 − 0.549i)3-s + (−0.404 + 0.293i)4-s + 0.447·5-s − 0.408·6-s + (−0.403 + 0.293i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (−0.170 + 0.124i)11-s + (0.0892 + 0.274i)12-s + (0.139 − 0.427i)13-s + (0.285 + 0.207i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.02 − 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129506 - 0.981984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129506 - 0.981984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + (-4.55 + 3.20i)T \) |
good | 7 | \( 1 + (1.06 - 0.775i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (0.567 - 0.412i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.501 + 1.54i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.24 + 3.08i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.71 + 5.28i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.350 + 0.254i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.62 + 8.06i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + (3.60 + 11.1i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 2.59i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.650 + 2.00i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.37 - 3.90i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.567 + 1.74i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 + (-0.612 - 0.445i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.635 - 0.461i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-8.87 - 6.44i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.02 - 6.22i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.32 + 3.14i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (10.7 - 7.80i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.575713649271958175516717868748, −9.025268018772969388232818767310, −8.178312481776472307831627382117, −7.15778204413609936662547354477, −6.33255519066521875961320008474, −5.27764999105674874024007606313, −4.15279072807109638610972918893, −2.77818164340780877627056217243, −2.17395415922432969077805703262, −0.47315144091856462591867351933,
1.74520675250748818091782825716, 3.33872315784734789879662748171, 4.31489158189081621483397818281, 5.30060410506893434923968312901, 6.28063562180372893205527833110, 6.89183738657484295979012832947, 8.130305595204003470014881860014, 8.733130169722296974218955181118, 9.550441874349665832096902496888, 10.34198921833032657270195635905