| L(s) = 1 | + (−0.831 + 2.55i)2-s + (−1.38 − 0.295i)3-s + (−4.23 − 3.07i)4-s + (−0.304 − 0.526i)5-s + (1.90 − 3.30i)6-s + (−1.57 + 0.702i)7-s + (7.04 − 5.11i)8-s + (−0.900 − 0.401i)9-s + (1.60 − 0.340i)10-s + (0.139 − 1.33i)11-s + (4.97 + 5.52i)12-s + (2.45 − 2.72i)13-s + (−0.485 − 4.62i)14-s + (0.266 + 0.821i)15-s + (4.00 + 12.3i)16-s + (0.288 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.587 + 1.80i)2-s + (−0.801 − 0.170i)3-s + (−2.11 − 1.53i)4-s + (−0.136 − 0.235i)5-s + (0.779 − 1.34i)6-s + (−0.596 + 0.265i)7-s + (2.49 − 1.80i)8-s + (−0.300 − 0.133i)9-s + (0.506 − 0.107i)10-s + (0.0421 − 0.401i)11-s + (1.43 + 1.59i)12-s + (0.681 − 0.756i)13-s + (−0.129 − 1.23i)14-s + (0.0688 + 0.212i)15-s + (1.00 + 3.07i)16-s + (0.0700 + 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00869602 + 0.353208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00869602 + 0.353208i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.831 - 2.55i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.38 + 0.295i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (0.304 + 0.526i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.57 - 0.702i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.139 + 1.33i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-2.45 + 2.72i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.288 - 2.74i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.71 + 1.90i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.436 + 0.316i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.51 - 7.73i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.101 + 0.0216i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.01 + 2.23i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (2.07 + 6.39i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.55 - 1.13i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.456 + 0.0969i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 1.93i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.784 - 7.46i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.01 - 9.64i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-16.3 + 3.46i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-12.3 - 9.00i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.03 - 0.751i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.33529954293179983720809764887, −9.150010752193533403213141835194, −8.624776671396156011557223962274, −7.964889613368457905584016321562, −6.70724062561119618140356290548, −6.39658465660661807764841412252, −5.55163462848977643660969080872, −4.92010571378005624900937627702, −3.48428990816806706277019065369, −0.913658607960050236145286282484,
0.31247548593109935308672818124, 1.82748159275620860052141187147, 3.03397043641906469183714704175, 3.94936188612689342134910589236, 4.82371584170105414862972490511, 6.05490655798255661348399463187, 7.25811120820037467485241815947, 8.304036410506398437801215209032, 9.230844737555443531460100255759, 9.780680482285410756937795048333
