L(s) = 1 | + (0.256 − 0.790i)3-s + (0.309 + 0.951i)4-s + (0.230 − 0.167i)5-s + (0.250 + 0.182i)9-s + 0.830·12-s + (−0.0730 − 0.224i)15-s + (−0.809 + 0.587i)16-s + (0.230 + 0.167i)20-s + 1.68·23-s + (−0.283 + 0.874i)25-s + (0.880 − 0.639i)27-s + (−1.36 − 0.988i)31-s + (−0.0957 + 0.294i)36-s + (−0.404 − 1.24i)37-s + 0.0881·45-s + ⋯ |
L(s) = 1 | + (0.256 − 0.790i)3-s + (0.309 + 0.951i)4-s + (0.230 − 0.167i)5-s + (0.250 + 0.182i)9-s + 0.830·12-s + (−0.0730 − 0.224i)15-s + (−0.809 + 0.587i)16-s + (0.230 + 0.167i)20-s + 1.68·23-s + (−0.283 + 0.874i)25-s + (0.880 − 0.639i)27-s + (−1.36 − 0.988i)31-s + (−0.0957 + 0.294i)36-s + (−0.404 − 1.24i)37-s + 0.0881·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.348444404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348444404\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - 1.68T + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 0.284T + T^{2} \) |
| 71 | \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.91T + T^{2} \) |
| 97 | \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.556018021178667258593969089825, −8.937578110224302910966690634135, −8.014380998298248514135363711225, −7.37003190035433347902823937327, −6.88862417761672350633697605215, −5.78225952331225686640665022401, −4.66944326717030601968764199555, −3.58405975131329532915825101491, −2.57872218986312447315675460325, −1.58511757954490734084886290531,
1.39127232227049611450809539498, 2.71515992428331541926843005493, 3.77735348047213346578201550901, 4.88652111160692234839603511548, 5.45092810105067994259763710340, 6.66999711282881372282782062010, 7.05056643738990634858083270620, 8.515572040745970818950576480451, 9.159264498606615825294635845272, 9.971684524123321734453882977136