L(s) = 1 | + (−1 − 1.73i)2-s − 3.04i·3-s + (−1.99 + 3.46i)4-s − 8.54·5-s + (−5.27 + 3.04i)6-s + 3.04i·7-s + 7.99·8-s − 0.274·9-s + (8.54 + 14.8i)10-s − 13.0i·11-s + (10.5 + 6.09i)12-s − 21.8·13-s + (5.27 − 3.04i)14-s + 26.0i·15-s + (−8 − 13.8i)16-s − 7.27·17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s − 1.01i·3-s + (−0.499 + 0.866i)4-s − 1.70·5-s + (−0.879 + 0.507i)6-s + 0.435i·7-s + 0.999·8-s − 0.0305·9-s + (0.854 + 1.48i)10-s − 1.18i·11-s + (0.879 + 0.507i)12-s − 1.67·13-s + (0.376 − 0.217i)14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s − 0.427·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0985210 + 0.367685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0985210 + 0.367685i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 + 3.04iT - 9T^{2} \) |
| 5 | \( 1 + 8.54T + 25T^{2} \) |
| 7 | \( 1 - 3.04iT - 49T^{2} \) |
| 11 | \( 1 + 13.0iT - 121T^{2} \) |
| 13 | \( 1 + 21.8T + 169T^{2} \) |
| 17 | \( 1 + 7.27T + 289T^{2} \) |
| 23 | \( 1 + 31.8iT - 529T^{2} \) |
| 29 | \( 1 - 4.37T + 841T^{2} \) |
| 31 | \( 1 - 21.0iT - 961T^{2} \) |
| 37 | \( 1 - 24.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 6.74T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 59.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 26.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 76.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 31.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 25.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 104. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 0.376iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 47.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.6T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14455781933853989774199308146, −12.18778460389666800233172683335, −11.77327076754771654675889425180, −10.55992759059793189461909514236, −8.807327855303875482613790456286, −7.923104400566152651706234610668, −7.01629702535989184296324053586, −4.46101652016326801070332098062, −2.76573626904382258291628180877, −0.38075786261024034835814460205,
4.12567214299558682290006844227, 4.85240841450964680616579909879, 7.21418394368749186378275950445, 7.71010912006590510594584597752, 9.345268806069075198547626287937, 10.15156645051961577363883938392, 11.37034496075491271213367983469, 12.66941284136967902796692738594, 14.49653989167092637913305017845, 15.26614631951368814647026235484