L(s) = 1 | + 29.7i·2-s − 81i·3-s − 373.·4-s + (343. − 1.35e3i)5-s + 2.40e3·6-s − 1.07e4i·7-s + 4.13e3i·8-s − 6.56e3·9-s + (4.03e4 + 1.02e4i)10-s + 5.44e4·11-s + 3.02e4i·12-s + 5.35e4i·13-s + 3.18e5·14-s + (−1.09e5 − 2.78e4i)15-s − 3.13e5·16-s − 6.44e5i·17-s + ⋯ |
L(s) = 1 | + 1.31i·2-s − 0.577i·3-s − 0.728·4-s + (0.245 − 0.969i)5-s + 0.759·6-s − 1.68i·7-s + 0.356i·8-s − 0.333·9-s + (1.27 + 0.323i)10-s + 1.12·11-s + 0.420i·12-s + 0.519i·13-s + 2.21·14-s + (−0.559 − 0.141i)15-s − 1.19·16-s − 1.87i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.67350 - 0.208833i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67350 - 0.208833i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81iT \) |
| 5 | \( 1 + (-343. + 1.35e3i)T \) |
good | 2 | \( 1 - 29.7iT - 512T^{2} \) |
| 7 | \( 1 + 1.07e4iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 5.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.35e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 6.44e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 2.10e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.52e4iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 2.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.48e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.19e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.48e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.80e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 5.85e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 5.05e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 5.84e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.07e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.46e7iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 4.06e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.31e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 8.40e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.88e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 1.04e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.28e9iT - 7.60e17T^{2} \) |
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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
−16.86235332210110757090495684642, −16.23017189538840069167717924047, −14.18084391123113425856583435825, −13.59008048512871170170068394800, −11.65232199251936106474116367116, −9.300652474075780121878158305361, −7.65697573113760645245414570357, −6.59052075004843036856490050916, −4.66401915695825710434697231049, −0.955486554575044734772895162382,
2.11630379973610380778822065997, 3.54232816679901321022532486156, 6.07471605351197105544272280434, 8.970064325986603370276539364579, 10.26238831714547390082896433568, 11.45393239954352505100904787589, 12.55435384820432006085869359678, 14.56382528058771620536915087587, 15.58306115817550856905887824665, 17.65943333752239688362778584499