| L(s) = 1 | + (−0.213 + 0.655i)2-s + (−0.882 − 0.187i)3-s + (1.23 + 0.896i)4-s + (−1.85 − 3.21i)5-s + (0.311 − 0.538i)6-s + (0.697 − 0.310i)7-s + (−1.96 + 1.42i)8-s + (−1.99 − 0.888i)9-s + (2.50 − 0.531i)10-s + (−0.430 + 4.09i)11-s + (−0.920 − 1.02i)12-s + (1.94 − 2.16i)13-s + (0.0550 + 0.523i)14-s + (1.03 + 3.18i)15-s + (0.424 + 1.30i)16-s + (0.136 + 1.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.463i)2-s + (−0.509 − 0.108i)3-s + (0.616 + 0.448i)4-s + (−0.829 − 1.43i)5-s + (0.127 − 0.219i)6-s + (0.263 − 0.117i)7-s + (−0.695 + 0.505i)8-s + (−0.665 − 0.296i)9-s + (0.790 − 0.168i)10-s + (−0.129 + 1.23i)11-s + (−0.265 − 0.295i)12-s + (0.540 − 0.599i)13-s + (0.0147 + 0.139i)14-s + (0.266 + 0.821i)15-s + (0.106 + 0.326i)16-s + (0.0331 + 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0882345 + 0.370073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0882345 + 0.370073i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.213 - 0.655i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.882 + 0.187i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (1.85 + 3.21i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.697 + 0.310i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.430 - 4.09i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.94 + 2.16i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.136 - 1.30i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (2.54 + 2.82i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.65 - 1.92i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.51 - 4.65i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (5.20 - 9.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.739 - 0.157i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.78 - 5.31i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (0.270 + 0.833i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.26 + 1.45i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.907 + 0.192i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 + (-1.04 - 1.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.06 + 3.14i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.591 - 5.62i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (1.47 + 14.0i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (13.7 - 2.93i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (3.58 + 2.60i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.26 - 3.10i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.53117413876852034460446468149, −9.220257112597176905906310900455, −8.486836785647406932477792007378, −7.952150946857498188120448529005, −7.10557581393604363739165875587, −6.13055225061426434723347859674, −5.20144083949587712911733588333, −4.38359075465988848537246623811, −3.16903577103952793279841216389, −1.51478448855232530386434619079,
0.19021757285398163163520083963, 2.16324586879743578749384746307, 3.11448969857785892851786483845, 4.00083598524924956651553318617, 5.66655404363495170938009551253, 6.16394103674685706875132288756, 6.99701302275305556235243746160, 7.978649398271740135060453933837, 8.822205905020855721367347805178, 10.13032583272504032417690836677
