| L(s) = 1 | + (−1.81 − 0.842i)2-s + (2.58 + 3.05i)4-s + 5.94i·5-s − 2.64i·7-s + (−2.10 − 7.71i)8-s + (5.00 − 10.7i)10-s − 15.0·11-s + 11.2i·13-s + (−2.22 + 4.79i)14-s + (−2.67 + 15.7i)16-s + 22.8·17-s − 33.0·19-s + (−18.1 + 15.3i)20-s + (27.3 + 12.7i)22-s + 1.69i·23-s + ⋯ |
| L(s) = 1 | + (−0.906 − 0.421i)2-s + (0.645 + 0.764i)4-s + 1.18i·5-s − 0.377i·7-s + (−0.263 − 0.964i)8-s + (0.500 − 1.07i)10-s − 1.37·11-s + 0.862i·13-s + (−0.159 + 0.342i)14-s + (−0.167 + 0.985i)16-s + 1.34·17-s − 1.74·19-s + (−0.907 + 0.766i)20-s + (1.24 + 0.577i)22-s + 0.0734i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.008627376871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008627376871\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.81 + 0.842i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 5.94iT - 25T^{2} \) |
| 11 | \( 1 + 15.0T + 121T^{2} \) |
| 13 | \( 1 - 11.2iT - 169T^{2} \) |
| 17 | \( 1 - 22.8T + 289T^{2} \) |
| 19 | \( 1 + 33.0T + 361T^{2} \) |
| 23 | \( 1 - 1.69iT - 529T^{2} \) |
| 29 | \( 1 + 28.9iT - 841T^{2} \) |
| 31 | \( 1 + 24.7iT - 961T^{2} \) |
| 37 | \( 1 + 53.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 30.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 44.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 72.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 96.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 86.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 36.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 13.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 66.0T + 9.40e3T^{2} \) |
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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
−10.36625446195061058726090836247, −9.709467222717727118082547709570, −8.439431524396542759622674620183, −7.65124623056958255555255202370, −6.91383863671848933645037418640, −5.93240709680919867557055602302, −4.16897725504119399107977099147, −3.00590188208518300477656677846, −2.04064431126330889687318773026, −0.00452975587301080532089693999,
1.44616035342177798343131104935, 2.93564869726786644769216667041, 4.97494133009753804691850255965, 5.42606727077850023786039265905, 6.61128800948336016226976508888, 7.998880741241585151471782936566, 8.263641514365959596759740556128, 9.153457596169261216301913569949, 10.26524917953105811975790863521, 10.65651259049847027199132917322
