| L(s) = 1 | + (1.73 − i)2-s + 1.73·3-s + (1.99 − 3.46i)4-s + (2.99 − 1.73i)6-s + 6.92·7-s − 7.99i·8-s + 2.99·9-s + 6.92i·11-s + (3.46 − 5.99i)12-s − 2i·13-s + (11.9 − 6.92i)14-s + (−8 − 13.8i)16-s + 10i·17-s + (5.19 − 2.99i)18-s − 20.7i·19-s + ⋯ |
| L(s) = 1 | + (0.866 − 0.5i)2-s + 0.577·3-s + (0.499 − 0.866i)4-s + (0.499 − 0.288i)6-s + 0.989·7-s − 0.999i·8-s + 0.333·9-s + 0.629i·11-s + (0.288 − 0.499i)12-s − 0.153i·13-s + (0.857 − 0.494i)14-s + (−0.5 − 0.866i)16-s + 0.588i·17-s + (0.288 − 0.166i)18-s − 1.09i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.04234 - 1.63693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.04234 - 1.63693i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 + i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6.92T + 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 2iT - 169T^{2} \) |
| 17 | \( 1 - 10iT - 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7T + 529T^{2} \) |
| 29 | \( 1 - 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2iT - 9.40e3T^{2} \) |
| 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
−11.51471262739585528595735283617, −10.56106718862072075090429517554, −9.726396628451234073194241805030, −8.507123415659137338826331701552, −7.46814456218183493527974991868, −6.30656011217088864318490102768, −4.95709854892453157190780095567, −4.18673373290333875589633645639, −2.74253590983432627857121751044, −1.55079743004161313379367721290,
1.99775882615400387046723522220, 3.42802634189928113137753482572, 4.50967753413399769410181074881, 5.59487103176334962874095512518, 6.72355855467881294539626396495, 8.023619955106863678175656637872, 8.265643624782410565805602507192, 9.722858645894143494450655798880, 11.03336149648084367239279054602, 11.79307541212737285826606307415
