L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.662 + 0.382i)3-s + (0.499 + 0.866i)4-s + (1.77 + 1.02i)5-s − 0.765·6-s + 0.999i·8-s + (−1.20 + 2.09i)9-s + (1.02 + 1.77i)10-s + (−3.14 − 1.06i)11-s + (−0.662 − 0.382i)12-s + 6.59·13-s − 1.57·15-s + (−0.5 + 0.866i)16-s + (1.80 + 3.12i)17-s + (−2.09 + 1.20i)18-s + (0.439 − 0.760i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.382 + 0.220i)3-s + (0.249 + 0.433i)4-s + (0.794 + 0.458i)5-s − 0.312·6-s + 0.353i·8-s + (−0.402 + 0.696i)9-s + (0.324 + 0.561i)10-s + (−0.947 − 0.320i)11-s + (−0.191 − 0.110i)12-s + 1.82·13-s − 0.405·15-s + (−0.125 + 0.216i)16-s + (0.437 + 0.758i)17-s + (−0.492 + 0.284i)18-s + (0.100 − 0.174i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.137622483\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.137622483\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.14 + 1.06i)T \) |
good | 3 | \( 1 + (0.662 - 0.382i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.77 - 1.02i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 + (-1.80 - 3.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.439 + 0.760i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.30 - 5.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.18iT - 29T^{2} \) |
| 31 | \( 1 + (5.61 - 3.24i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.52 + 9.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 - 7.81iT - 43T^{2} \) |
| 47 | \( 1 + (-4.97 - 2.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.214 + 0.372i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.78 - 1.60i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.66 + 9.81i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.48 + 2.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 + (-5.41 - 9.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + (5.82 + 3.36i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
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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
−10.49596096722305249682846470986, −9.306672169524564665839882100525, −8.295567276861996039196036452268, −7.65538186558080485826710414213, −6.42151663883699400155515552104, −5.72813064738181180984791667582, −5.36616774383213937194036026896, −3.97490854024293319983405691875, −3.04386536272703912611255228772, −1.79453109743576569174745342190,
0.828850451186696781327976153563, 2.11852926680351253583531465417, 3.31253369441316338201808278224, 4.38166349523511581796462513000, 5.60356366254043525737724069516, 5.86164004647448628603522219382, 6.79758519993795739189240384338, 8.029964278743761704923420383545, 8.910574155915376617118927476839, 9.774080148462216776842050900712