L(s) = 1 | + (−1.21 − 0.729i)2-s + (−3.10 − 1.08i)3-s + (0.934 + 1.76i)4-s + (−1.05 − 0.241i)5-s + (2.96 + 3.58i)6-s + (1.22 + 2.54i)7-s + (0.158 − 2.82i)8-s + (6.11 + 4.87i)9-s + (1.10 + 1.06i)10-s + (−0.0781 + 0.693i)11-s + (−0.980 − 6.50i)12-s + (−2.28 + 1.82i)13-s + (0.372 − 3.97i)14-s + (3.01 + 1.89i)15-s + (−2.25 + 3.30i)16-s + (−3.47 + 3.47i)17-s + ⋯ |
L(s) = 1 | + (−0.856 − 0.516i)2-s + (−1.79 − 0.627i)3-s + (0.467 + 0.884i)4-s + (−0.472 − 0.107i)5-s + (1.21 + 1.46i)6-s + (0.463 + 0.961i)7-s + (0.0560 − 0.998i)8-s + (2.03 + 1.62i)9-s + (0.349 + 0.336i)10-s + (−0.0235 + 0.209i)11-s + (−0.283 − 1.87i)12-s + (−0.634 + 0.506i)13-s + (0.0996 − 1.06i)14-s + (0.779 + 0.489i)15-s + (−0.563 + 0.826i)16-s + (−0.843 + 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.226114 + 0.136792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.226114 + 0.136792i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.21 + 0.729i)T \) |
| 29 | \( 1 + (2.03 - 4.98i)T \) |
good | 3 | \( 1 + (3.10 + 1.08i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (1.05 + 0.241i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 2.54i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (0.0781 - 0.693i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (2.28 - 1.82i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (3.47 - 3.47i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.15 - 3.31i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (-2.55 + 0.582i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (1.98 - 1.24i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (-9.64 + 1.08i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-0.357 - 0.357i)T + 41iT^{2} \) |
| 43 | \( 1 + (5.65 - 8.99i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (4.77 + 0.538i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (0.388 - 1.70i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 4.97iT - 59T^{2} \) |
| 61 | \( 1 + (-2.11 + 6.04i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (-2.96 + 3.71i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (4.27 + 5.36i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (3.78 - 6.01i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (1.34 - 0.151i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (1.23 - 2.55i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (2.75 + 4.37i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (-0.776 - 2.21i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−12.95769932055520923248200062385, −12.32858623915870025664849313224, −11.56365908921423395458488749231, −10.98757406873536498156676297782, −9.747951566201731020032883392025, −8.251277493476358300001968586884, −7.17139491456634460422655630571, −6.03665589149249538315215639926, −4.58004172343938214477660520996, −1.80090391462244341718682542432,
0.47371697609282356396132075276, 4.45722015432302233914769050002, 5.46359840717920257397438826318, 6.80036848209171667176522968709, 7.59146379098894027434472885817, 9.430614828190463112712805121417, 10.32546110907147686463102974570, 11.28241485560608923859539198271, 11.58880384901774542533727858347, 13.32428615803022581821272482928