| L(s) = 1 | + (−0.978 − 0.207i)2-s + (0.0943 − 1.72i)3-s + (0.913 + 0.406i)4-s + (1.33 − 1.79i)5-s + (−0.451 + 1.67i)6-s + (−2.20 + 3.81i)7-s + (−0.809 − 0.587i)8-s + (−2.98 − 0.326i)9-s + (−1.68 + 1.47i)10-s + (−6.16 − 1.30i)11-s + (0.789 − 1.54i)12-s + (−2.72 + 0.580i)13-s + (2.94 − 3.27i)14-s + (−2.97 − 2.48i)15-s + (0.669 + 0.743i)16-s + (2.99 + 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.0544 − 0.998i)3-s + (0.456 + 0.203i)4-s + (0.598 − 0.801i)5-s + (−0.184 + 0.682i)6-s + (−0.832 + 1.44i)7-s + (−0.286 − 0.207i)8-s + (−0.994 − 0.108i)9-s + (−0.531 + 0.466i)10-s + (−1.85 − 0.394i)11-s + (0.227 − 0.445i)12-s + (−0.756 + 0.160i)13-s + (0.787 − 0.874i)14-s + (−0.767 − 0.640i)15-s + (0.167 + 0.185i)16-s + (0.726 + 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0556465 + 0.239827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0556465 + 0.239827i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (-0.0943 + 1.72i)T \) |
| 5 | \( 1 + (-1.33 + 1.79i)T \) |
| good | 7 | \( 1 + (2.20 - 3.81i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (6.16 + 1.30i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.72 - 0.580i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 2.17i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.85 + 4.25i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.52 + 1.68i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.333 + 3.16i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.269 + 2.56i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (2.50 - 7.69i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.59 - 0.338i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.37 + 2.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.591 + 5.63i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-8.11 + 5.89i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.16 - 0.884i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-2.16 - 0.460i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.101 + 0.963i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (0.214 - 0.156i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.82 - 8.68i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 14.5i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (2.29 - 1.02i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.32 + 4.08i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.494 + 4.70i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.39751917609505830233481095576, −9.562778032081345564884922644016, −8.509154367992955548483799765135, −8.273641836880513601306078487082, −6.87868075901402685371981513637, −5.90432756850179221770545693293, −5.21351684812477153668589148763, −2.74522453745087348845470116609, −2.20907941737545511414785457486, −0.16879739060894641080825135821,
2.53176485662274879361933962909, 3.53282625642443290742992776858, 4.99449538772035646100169392747, 6.03721077678499908898795227759, 7.24050664949591641664467078783, 7.80469656783074842742541259305, 9.249766966543699608558481813986, 10.11421597835493627620990127610, 10.41768673097992685353225749152, 10.85211069664280152514409767003
