| L(s) = 1 | + (−0.900 + 0.433i)2-s + (−1.94 + 1.32i)3-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (1.17 − 2.03i)6-s + (1.33 + 2.30i)7-s + (−0.222 + 0.974i)8-s + (0.923 − 2.35i)9-s + (0.955 + 0.294i)10-s + (2.03 + 2.55i)11-s + (−0.175 + 2.34i)12-s + (−2.20 + 0.679i)13-s + (−2.19 − 1.49i)14-s + (2.32 + 0.350i)15-s + (−0.222 − 0.974i)16-s + (−3.42 + 3.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 + 0.306i)2-s + (−1.12 + 0.764i)3-s + (0.311 − 0.390i)4-s + (−0.327 − 0.304i)5-s + (0.479 − 0.831i)6-s + (0.503 + 0.871i)7-s + (−0.0786 + 0.344i)8-s + (0.307 − 0.784i)9-s + (0.302 + 0.0932i)10-s + (0.613 + 0.769i)11-s + (−0.0507 + 0.676i)12-s + (−0.610 + 0.188i)13-s + (−0.587 − 0.400i)14-s + (0.600 + 0.0904i)15-s + (−0.0556 − 0.243i)16-s + (−0.829 + 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0565137 - 0.248481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0565137 - 0.248481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 - 0.433i)T \) |
| 5 | \( 1 + (0.733 + 0.680i)T \) |
| 43 | \( 1 + (-0.0655 + 6.55i)T \) |
| good | 3 | \( 1 + (1.94 - 1.32i)T + (1.09 - 2.79i)T^{2} \) |
| 7 | \( 1 + (-1.33 - 2.30i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.03 - 2.55i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.20 - 0.679i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (3.42 - 3.17i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.33 + 3.41i)T + (-13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (0.878 - 0.132i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (5.38 + 3.67i)T + (10.5 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.0946 + 1.26i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + (-0.861 + 1.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.53 - 4.10i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (5.86 - 7.35i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.72 + 2.07i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.719 - 3.15i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.202 - 2.70i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-2.88 - 7.35i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (5.36 + 0.808i)T + (67.8 + 20.9i)T^{2} \) |
| 73 | \( 1 + (7.51 - 2.31i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 2.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.4 + 7.11i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (14.1 - 9.68i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (-11.8 - 14.8i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.57202639911826471737257523856, −10.81334103518319453263492814075, −9.815761622026660527581421390922, −9.095740329579214504600988682556, −8.163258732162306170596157138437, −6.94474755710423845351702589898, −5.98063542523396498287940424804, −5.00650549318380323308575872776, −4.25899768550981215659602812234, −2.02797176551937813716825816752,
0.22771834775728110562598797127, 1.62426312723427536196968562846, 3.47872268996322746359657015465, 4.86605267109414970455186797628, 6.19644212752044240520658780158, 7.00219899373893247861429948228, 7.68414469346965185862460192498, 8.759272294132959750628025412259, 9.996150488060743938447509260111, 10.95074452085618804449248749647
