L(s) = 1 | − 5.65i·2-s − 13.8i·3-s − 32.0·4-s − 9.64·5-s − 78.3·6-s − 472.·7-s + 181. i·8-s + 537.·9-s + 54.5i·10-s − 1.63e3·11-s + 443. i·12-s + 2.50e3i·13-s + 2.67e3i·14-s + 133. i·15-s + 1.02e3·16-s + 1.38e3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.513i·3-s − 0.500·4-s − 0.0771·5-s − 0.362·6-s − 1.37·7-s + 0.353i·8-s + 0.736·9-s + 0.0545i·10-s − 1.22·11-s + 0.256i·12-s + 1.14i·13-s + 0.974i·14-s + 0.0396i·15-s + 0.250·16-s + 0.282·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0526066 + 0.0921067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0526066 + 0.0921067i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 5.65iT \) |
| 19 | \( 1 + (5.90e3 - 3.48e3i)T \) |
good | 3 | \( 1 + 13.8iT - 729T^{2} \) |
| 5 | \( 1 + 9.64T + 1.56e4T^{2} \) |
| 7 | \( 1 + 472.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.63e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.50e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 1.38e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 4.75e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.29e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 7.23e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 9.60e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 2.72e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.97e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.84e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.30e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.12e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.43e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.23e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.81e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.39e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.67e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 8.38e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 8.65e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.02e6iT - 8.32e11T^{2} \) |
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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
−13.74177357495014286804170458814, −12.93687524240142153782897660265, −12.05509291035697622297353363176, −10.41553993573649166926916547749, −9.474305364344065934198794555980, −7.71264107826847770749853497724, −6.19023133993537112772124172249, −4.02479848563902741383354675949, −2.20031191778455532407894335979, −0.04982917952764402625666144158,
3.34342810804552054452169217127, 5.12486526502529185024049109391, 6.64386407791393237094142261135, 8.078152903437930763102126797724, 9.707517327203781170010945021123, 10.46354757572892532992475630982, 12.70999228735663540688160891657, 13.29578662207791866039911517938, 15.12886906024486088723825812990, 15.75546773064510961300323448073