L(s) = 1 | + (−0.309 + 0.951i)2-s + (1.98 + 1.44i)3-s + (−0.809 − 0.587i)4-s + (1.96 − 1.06i)5-s + (−1.98 + 1.44i)6-s − 3.97·7-s + (0.809 − 0.587i)8-s + (0.935 + 2.87i)9-s + (0.402 + 2.19i)10-s + (−1.61 + 4.96i)11-s + (−0.758 − 2.33i)12-s + (1.42 + 4.39i)13-s + (1.22 − 3.77i)14-s + (5.44 + 0.729i)15-s + (0.309 + 0.951i)16-s + (−1.21 + 0.880i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (1.14 + 0.833i)3-s + (−0.404 − 0.293i)4-s + (0.879 − 0.475i)5-s + (−0.810 + 0.589i)6-s − 1.50·7-s + (0.286 − 0.207i)8-s + (0.311 + 0.959i)9-s + (0.127 + 0.695i)10-s + (−0.486 + 1.49i)11-s + (−0.218 − 0.673i)12-s + (0.395 + 1.21i)13-s + (0.328 − 1.00i)14-s + (1.40 + 0.188i)15-s + (0.0772 + 0.237i)16-s + (−0.293 + 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.552745 + 1.65784i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.552745 + 1.65784i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-1.98 - 1.44i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 3.97T + 7T^{2} \) |
| 11 | \( 1 + (1.61 - 4.96i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 4.39i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.21 - 0.880i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 + (-0.570 + 1.75i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.42 - 3.93i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.35 - 3.89i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.33 - 10.2i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.117 - 0.361i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + (3.14 + 2.28i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.12 + 4.45i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.60 + 4.95i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 5.74i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (10.1 - 7.36i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.06 - 2.22i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.68 + 14.4i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-11.5 - 8.37i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.42 + 5.39i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.888 + 2.73i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (6.37 + 4.63i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.964608832609194438630671727379, −9.355982109469680737836611038893, −9.027162553848941301232432974191, −8.093032380577336227574007059716, −6.79989966137360168739691930964, −6.38276908125962263033698544442, −4.96758899315907402604367513806, −4.27393754901562391358453970631, −3.09867131355146430438831680925, −1.93952971763766451367502333429,
0.74484368147573313696970785092, 2.41998659192132180401319227924, 2.93634846487503656489109033536, 3.61896903737880805690293365910, 5.67281763376174106221801215043, 6.25509743462989543774514485322, 7.38066183543473404463283511564, 8.137384393795673949694272575214, 9.086709729326178245398254842498, 9.487833312226520821763472755461