| L(s) = 1 | + (−1.90 + 0.618i)2-s + (3.23 − 2.35i)4-s − 5.25·7-s + (−4.70 + 6.47i)8-s − 19.9i·11-s − 8.47i·13-s + (9.99 − 3.24i)14-s + (4.94 − 15.2i)16-s + 11.8i·17-s + 15.2i·19-s + (12.3 + 37.8i)22-s − 0.555·23-s + (5.23 + 16.1i)26-s + (−17.0 + 12.3i)28-s − 10.9·29-s + ⋯ |
| L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s − 0.751·7-s + (−0.587 + 0.809i)8-s − 1.81i·11-s − 0.651i·13-s + (0.714 − 0.232i)14-s + (0.309 − 0.951i)16-s + 0.699i·17-s + 0.800i·19-s + (0.559 + 1.72i)22-s − 0.0241·23-s + (0.201 + 0.619i)26-s + (−0.607 + 0.441i)28-s − 0.377·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1506933712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1506933712\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.90 - 0.618i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.25T + 49T^{2} \) |
| 11 | \( 1 + 19.9iT - 121T^{2} \) |
| 13 | \( 1 + 8.47iT - 169T^{2} \) |
| 17 | \( 1 - 11.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.555T + 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 + 8.29iT - 961T^{2} \) |
| 37 | \( 1 - 18.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 76.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.8iT - 9.40e3T^{2} \) |
| 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
−10.15002509276289149534336219894, −9.439685393121704290947992931410, −8.382648298580394236018355370466, −8.104292522220335881232683660531, −6.85061374333425360088372297084, −6.03623197813838201950003303455, −5.50785471890458004652452212216, −3.67798752104931594096297049205, −2.78683226327264713746826759168, −1.18390166086736544379095978964,
0.07171840508930801372351664917, 1.74671266021176587676445761238, 2.72029458406553535689003061424, 3.94533233315345884430363816723, 5.07714398849144474483821175083, 6.66067734021638596318433062305, 6.94204613470987794761238512189, 7.88391396907213482778967845605, 9.000506643917526391905884913193, 9.646877890498759332984529609135
