| 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
−9.425384894055397052338868966292, −8.966627070208442104990042412233, −7.960523971339126951096432844677, −7.57977888105580320184521300627, −6.64112089277006000786868499735, −4.85754272827958117388722276158, −3.99799001830553335049153678300, −2.50829320330972878203272048418, −1.58288930365673033583250952737, −0.44321970753655812902979088555,
1.35401583508817817880474944549, 3.08973351906036010212510187301, 4.50775897233117956977023198114, 5.71966578762489626284969062299, 6.38217115222478244491464976243, 7.52109388612304953281676079786, 7.77461841089665047211829881019, 8.727534271194196307794537909443, 9.495311327236826850446705095836, 10.52740309089665797513994122462
