| L(s) = 1 | − 40.3i·3-s − 186.·5-s + 202.·7-s − 898.·9-s − 1.59e3·11-s − 143. i·13-s + 7.53e3i·15-s + 4.77e3·17-s + (−1.01e3 + 6.78e3i)19-s − 8.16e3i·21-s + 7.73e3·23-s + 1.92e4·25-s + 6.83e3i·27-s − 8.82e3i·29-s + 5.15e4i·31-s + ⋯ |
| L(s) = 1 | − 1.49i·3-s − 1.49·5-s + 0.589·7-s − 1.23·9-s − 1.19·11-s − 0.0652i·13-s + 2.23i·15-s + 0.971·17-s + (−0.147 + 0.989i)19-s − 0.881i·21-s + 0.635·23-s + 1.23·25-s + 0.347i·27-s − 0.361i·29-s + 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.187357 + 0.161518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187357 + 0.161518i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.01e3 - 6.78e3i)T \) |
| good | 3 | \( 1 + 40.3iT - 729T^{2} \) |
| 5 | \( 1 + 186.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 202.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.59e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 143. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.77e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 7.73e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 8.82e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.15e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 9.46e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.87e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.29e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 9.72e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 4.61e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 6.17e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.56e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.90e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.08e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.30e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.06e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.22e4T + 3.26e11T^{2} \) |
| 89 | \( 1 - 5.61e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.51e5iT - 8.32e11T^{2} \) |
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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
−13.33433679934479817290468659184, −12.30864949331904822117759011145, −11.76380754910992687093295778237, −10.50154751867358037301088884852, −8.158683876762347505190363806333, −7.961642605726728131620105117031, −6.81579948605464028420962297232, −5.08627815986691194357724860715, −3.19618052833142988101970536198, −1.37457836824521052146153700663,
0.099918861017052233589202121178, 3.14135421653420517638554082514, 4.33644624358380704148085202061, 5.22457789001735216481205751997, 7.51353674409338609420878058380, 8.445897983097858856999705937342, 9.785941567124836651944950121113, 10.96172415365668557215206806695, 11.49515246172940547217160330113, 12.96353956266022525185840016262
