L(s) = 1 | + (4.21 + 3.04i)3-s − 9.33·5-s − 36.3i·7-s + (8.50 + 25.6i)9-s − 48.4i·11-s + 25.8i·13-s + (−39.3 − 28.3i)15-s + 74.2i·17-s + 82.9·19-s + (110. − 153. i)21-s − 179.·23-s − 37.8·25-s + (−42.1 + 133. i)27-s − 122.·29-s − 64.1i·31-s + ⋯ |
L(s) = 1 | + (0.810 + 0.585i)3-s − 0.834·5-s − 1.96i·7-s + (0.314 + 0.949i)9-s − 1.32i·11-s + 0.552i·13-s + (−0.676 − 0.488i)15-s + 1.05i·17-s + 1.00·19-s + (1.14 − 1.59i)21-s − 1.62·23-s − 0.303·25-s + (−0.300 + 0.953i)27-s − 0.783·29-s − 0.371i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3246780742\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3246780742\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.21 - 3.04i)T \) |
good | 5 | \( 1 + 9.33T + 125T^{2} \) |
| 7 | \( 1 + 36.3iT - 343T^{2} \) |
| 11 | \( 1 + 48.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 25.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 74.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 82.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 5.01iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 325. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 95.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 185.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 226. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 198. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 23.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 669.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 229. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 321. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 131. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 136.T + 9.12e5T^{2} \) |
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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
−9.701454224824636261285376023307, −8.435700091567816030519358132264, −7.908671611205851301077802781407, −7.25022539241886940222469396694, −6.01960847075378820738445483154, −4.52698666160180634615451009709, −3.81829993042165840265039403653, −3.37561018996525253189410955435, −1.51074460813933017992356776137, −0.07508105710161636313119954100,
1.83382173446204280374501024847, 2.64882518252887458581467759238, 3.67698157652848125419849524352, 5.00349748011227719832669355167, 5.93761929203831036500828923005, 7.15782138072906672285286684559, 7.78434531783814475056195465229, 8.557757250046638534069552282273, 9.372977083949229860854610380279, 9.966923877204743571228398056317