L(s) = 1 | + 3.44i·2-s − 7.87·4-s + 6.27i·5-s + 8.87·7-s − 13.3i·8-s − 21.6·10-s + 24.7·13-s + 30.5i·14-s + 14.4·16-s + 4.68i·17-s − 29.2·19-s − 49.3i·20-s + 27.7i·23-s − 14.3·25-s + 85.2i·26-s + ⋯ |
L(s) = 1 | + 1.72i·2-s − 1.96·4-s + 1.25i·5-s + 1.26·7-s − 1.66i·8-s − 2.16·10-s + 1.90·13-s + 2.18i·14-s + 0.905·16-s + 0.275i·17-s − 1.53·19-s − 2.46i·20-s + 1.20i·23-s − 0.574·25-s + 3.27i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.845955139\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.845955139\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 3.44iT - 4T^{2} \) |
| 5 | \( 1 - 6.27iT - 25T^{2} \) |
| 7 | \( 1 - 8.87T + 49T^{2} \) |
| 13 | \( 1 - 24.7T + 169T^{2} \) |
| 17 | \( 1 - 4.68iT - 289T^{2} \) |
| 19 | \( 1 + 29.2T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 34.5iT - 841T^{2} \) |
| 31 | \( 1 - 24.7T + 961T^{2} \) |
| 37 | \( 1 + 1.36T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 23.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.32iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 36.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.12T + 4.48e3T^{2} \) |
| 71 | \( 1 + 33.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.76T + 5.32e3T^{2} \) |
| 79 | \( 1 - 3.01T + 6.24e3T^{2} \) |
| 83 | \( 1 - 43.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 105. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 102.T + 9.40e3T^{2} \) |
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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
−10.15115894797563961886532442591, −8.870756846399824506799458960738, −8.303649729218058543913191451942, −7.73320904533512328664626593007, −6.64749642883988668053022054011, −6.34545364721885904960781462803, −5.35005389266830876238299897617, −4.37971546164438833898447109328, −3.37785383047930865819188910154, −1.61569073827847634773787381173,
0.61807720387855055469328090350, 1.45906789195658397660211159000, 2.34336952943031546766718937945, 3.88784482737672385903390651281, 4.41228157826053193129738418023, 5.19890265298148709711659441656, 6.39443442331013067554235241733, 8.234079122116888924389010419508, 8.501096918329505553754520317930, 9.075986414602495046554614860463