L(s) = 1 | + (−1 + i)2-s + (−0.866 − 1.5i)3-s − 2i·4-s + (−3 − 1.73i)5-s + (2.36 + 0.633i)6-s + (2.59 + 0.5i)7-s + (2 + 2i)8-s + (−1.5 + 2.59i)9-s + (4.73 − 1.26i)10-s + (−1.73 + i)11-s + (−3 + 1.73i)12-s + (−4.5 + 2.59i)13-s + (−3.09 + 2.09i)14-s + 6i·15-s − 4·16-s + (−4.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.499 − 0.866i)3-s − i·4-s + (−1.34 − 0.774i)5-s + (0.965 + 0.258i)6-s + (0.981 + 0.188i)7-s + (0.707 + 0.707i)8-s + (−0.5 + 0.866i)9-s + (1.49 − 0.400i)10-s + (−0.522 + 0.301i)11-s + (−0.866 + 0.499i)12-s + (−1.24 + 0.720i)13-s + (−0.827 + 0.560i)14-s + 1.54i·15-s − 16-s + (−1.09 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 7 | \( 1 + (-2.59 - 0.5i)T \) |
good | 5 | \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 + 5.19iT - 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 3iT - 79T^{2} \) |
| 83 | \( 1 + (-2.59 + 4.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.55267936016284721554525320053, −10.81527602676702048522004998949, −9.201777011342048515482415272553, −8.356146977247064240702626260333, −7.48152746099534992270864424385, −7.03293301250601471266251954068, −5.21971357394230672962812450717, −4.74205698022061536421782953276, −1.84937841768072233237056378663, 0,
2.81488133743212904072203226784, 4.00474979300844267647547283737, 4.96636417586955352086330618606, 6.95287434774973908929836873107, 7.86387797654245248854361540894, 8.695252479940453419913592429805, 10.04938985228442824038640900903, 10.81735978358261038575159264731, 11.31351647326737912339660476552