| L(s) = 1 | + (−2.11 + 1.53i)2-s + (−0.0913 + 0.869i)3-s + (1.5 − 4.61i)4-s + (−1.11 + 1.93i)5-s + (−1.14 − 1.98i)6-s + (0.978 − 0.207i)7-s + (2.30 + 7.10i)8-s + (2.18 + 0.464i)9-s + (−0.611 − 5.82i)10-s + (−2.83 − 3.15i)11-s + (3.87 + 1.72i)12-s + (2.39 − 1.06i)13-s + (−1.75 + 1.94i)14-s + (−1.58 − 1.14i)15-s + (−7.97 − 5.79i)16-s + (−2.47 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 + 1.08i)2-s + (−0.0527 + 0.501i)3-s + (0.750 − 2.30i)4-s + (−0.499 + 0.866i)5-s + (−0.467 − 0.809i)6-s + (0.369 − 0.0785i)7-s + (0.816 + 2.51i)8-s + (0.729 + 0.154i)9-s + (−0.193 − 1.84i)10-s + (−0.855 − 0.950i)11-s + (1.11 + 0.498i)12-s + (0.664 − 0.295i)13-s + (−0.468 + 0.519i)14-s + (−0.408 − 0.296i)15-s + (−1.99 − 1.44i)16-s + (−0.600 + 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0604993 - 0.513235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0604993 - 0.513235i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.11 - 1.53i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.0913 - 0.869i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.978 + 0.207i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (2.83 + 3.15i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.39 + 1.06i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (2.47 - 2.75i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.913 - 0.406i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.810 - 2.49i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.437 - 0.317i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.12 - 3.67i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.153 - 1.46i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-8.85 - 3.94i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-7.85 - 5.70i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (13.4 + 2.85i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (1.24 - 11.8i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.43 - 0.306i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (2.83 + 3.15i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-1.08 + 1.20i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.330 + 3.14i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.74 + 14.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.16 - 6.65i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.52740309089665797513994122462, −9.495311327236826850446705095836, −8.727534271194196307794537909443, −7.77461841089665047211829881019, −7.52109388612304953281676079786, −6.38217115222478244491464976243, −5.71966578762489626284969062299, −4.50775897233117956977023198114, −3.08973351906036010212510187301, −1.35401583508817817880474944549,
0.44321970753655812902979088555, 1.58288930365673033583250952737, 2.50829320330972878203272048418, 3.99799001830553335049153678300, 4.85754272827958117388722276158, 6.64112089277006000786868499735, 7.57977888105580320184521300627, 7.960523971339126951096432844677, 8.966627070208442104990042412233, 9.425384894055397052338868966292
