| 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
−9.780680482285410756937795048333, −9.230844737555443531460100255759, −8.304036410506398437801215209032, −7.25811120820037467485241815947, −6.05490655798255661348399463187, −4.82371584170105414862972490511, −3.94936188612689342134910589236, −3.03397043641906469183714704175, −1.82748159275620860052141187147, −0.31247548593109935308672818124,
0.913658607960050236145286282484, 3.48428990816806706277019065369, 4.92010571378005624900937627702, 5.55163462848977643660969080872, 6.39658465660661807764841412252, 6.70724062561119618140356290548, 7.964889613368457905584016321562, 8.624776671396156011557223962274, 9.150010752193533403213141835194, 10.33529954293179983720809764887
