L(s) = 1 | + (0.230 − 0.167i)3-s + (−0.809 − 0.587i)4-s + (0.519 + 1.60i)5-s + (−0.283 + 0.874i)9-s − 0.284·12-s + (0.387 + 0.281i)15-s + (0.309 + 0.951i)16-s + (0.519 − 1.60i)20-s − 1.30·23-s + (−1.48 + 1.07i)25-s + (0.168 + 0.519i)27-s + (−0.404 + 1.24i)31-s + (0.743 − 0.540i)36-s + (1.55 + 1.12i)37-s − 1.54·45-s + ⋯ |
L(s) = 1 | + (0.230 − 0.167i)3-s + (−0.809 − 0.587i)4-s + (0.519 + 1.60i)5-s + (−0.283 + 0.874i)9-s − 0.284·12-s + (0.387 + 0.281i)15-s + (0.309 + 0.951i)16-s + (0.519 − 1.60i)20-s − 1.30·23-s + (−1.48 + 1.07i)25-s + (0.168 + 0.519i)27-s + (−0.404 + 1.24i)31-s + (0.743 − 0.540i)36-s + (1.55 + 1.12i)37-s − 1.54·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9416692304\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9416692304\) |
\(L(1)\) |
|
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 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.519 - 1.60i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + 1.30T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-1.55 + 1.12i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 1.68T + T^{2} \) |
| 71 | \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 0.830T + T^{2} \) |
| 97 | \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.08899718265880167279658574412, −9.330698195144209155717716230788, −8.325704186557341239516357118220, −7.58331224739838081687275792073, −6.59528893621422752254406777065, −5.90767006580290715741230101976, −5.05329710071332257414607160238, −3.87973814056114463076564936873, −2.79370040319787209309081043676, −1.84546903389970258791212010733,
0.812435283290659250667624295587, 2.44857591627705761634283245239, 3.96717275352085414746820209330, 4.28691464047454845530236392244, 5.48025291801082192548156885669, 6.02429968551097391027111287938, 7.54761106146349996957922205854, 8.267984125555782838457960156937, 8.969098448696220097160468355188, 9.412397443440432357841062313997