L(s) = 1 | + (1 − i)2-s + (−1.94 + 2.28i)3-s − 2i·4-s + (4.58 + 1.99i)5-s + (0.347 + 4.22i)6-s + (−4.23 + 5.57i)7-s + (−2 − 2i)8-s + (−1.46 − 8.87i)9-s + (6.58 − 2.58i)10-s + 14.6i·11-s + (4.57 + 3.88i)12-s + (3.48 + 3.48i)13-s + (1.34 + 9.80i)14-s + (−13.4 + 6.61i)15-s − 4·16-s + (20.1 + 20.1i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (−0.646 + 0.762i)3-s − 0.5i·4-s + (0.916 + 0.399i)5-s + (0.0579 + 0.704i)6-s + (−0.604 + 0.796i)7-s + (−0.250 − 0.250i)8-s + (−0.163 − 0.986i)9-s + (0.658 − 0.258i)10-s + 1.32i·11-s + (0.381 + 0.323i)12-s + (0.267 + 0.267i)13-s + (0.0961 + 0.700i)14-s + (−0.897 + 0.441i)15-s − 0.250·16-s + (1.18 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.35385 + 0.874285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35385 + 0.874285i\) |
\(L(2)\) |
|
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 | \( 1 + (1.94 - 2.28i)T \) |
| 5 | \( 1 + (-4.58 - 1.99i)T \) |
| 7 | \( 1 + (4.23 - 5.57i)T \) |
good | 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + (-3.48 - 3.48i)T + 169iT^{2} \) |
| 17 | \( 1 + (-20.1 - 20.1i)T + 289iT^{2} \) |
| 19 | \( 1 + 26.4T + 361T^{2} \) |
| 23 | \( 1 + (-2.68 - 2.68i)T + 529iT^{2} \) |
| 29 | \( 1 - 28.5T + 841T^{2} \) |
| 31 | \( 1 + 15.6iT - 961T^{2} \) |
| 37 | \( 1 + (-7.69 - 7.69i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 37.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-41.7 + 41.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (21.0 + 21.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (47.4 + 47.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 61.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-68.9 - 68.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 65.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (6.51 + 6.51i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 42.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (9.52 - 9.52i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 19.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-84.6 + 84.6i)T - 9.40e3iT^{2} \) |
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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
−12.47110099196454547828510804020, −11.33610642713000888604166567057, −10.11233585966696093273203735428, −9.982299861991150681467796120393, −8.756447863533085723656142631424, −6.63307536017597901209272607520, −5.96756489661317196538279750469, −4.90463472810663586199251331345, −3.57409390309597827888446891131, −2.05323603106399761578537988229,
0.862105086905449307468963163199, 2.98087177759059170112243004585, 4.78821149334363491023525098358, 5.93282116868229097882691667543, 6.48414573475529633490574611095, 7.69321706480487449472905080446, 8.778097239391300553246663518755, 10.17670684547505022289251536952, 11.07326030054244688439102923468, 12.32163191896199096228033805415