L(s) = 1 | + (−0.395 − 0.228i)2-s + 3-s + (−0.895 − 1.55i)4-s + (−1.5 + 0.866i)5-s + (−0.395 − 0.228i)6-s − 2.64i·7-s + 1.73i·8-s + 9-s + 0.791·10-s − 3.46i·11-s + (−0.895 − 1.55i)12-s + (−1 − 3.46i)13-s + (−0.604 + 1.04i)14-s + (−1.5 + 0.866i)15-s + (−1.39 + 2.41i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.279 − 0.161i)2-s + 0.577·3-s + (−0.447 − 0.775i)4-s + (−0.670 + 0.387i)5-s + (−0.161 − 0.0932i)6-s − 0.999i·7-s + 0.612i·8-s + 0.333·9-s + 0.250·10-s − 1.04i·11-s + (−0.258 − 0.447i)12-s + (−0.277 − 0.960i)13-s + (−0.161 + 0.279i)14-s + (−0.387 + 0.223i)15-s + (−0.348 + 0.604i)16-s + (−0.121 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.544920 - 0.753660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.544920 - 0.753660i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + 2.64iT \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (0.395 + 0.228i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 + (0.291 - 0.504i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.708 + 0.409i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.08 - 1.77i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.08 + 3.51i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.29 + 3.96i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.29 + 3.05i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.08 - 10.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.29 - 4.78i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.11iT - 67T^{2} \) |
| 71 | \( 1 + (-9.87 - 5.70i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.29 - 9.16i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0825 + 0.0476i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.09221195693291006099442353488, −10.82495371950902107498224065973, −9.736856686241309379605988488294, −8.774886395774201195497950970186, −7.84080404730479738269794113401, −6.87913316312645328137726413251, −5.42127565260545621576359315806, −4.15888800838187748054410391102, −2.93837800080755273840951834984, −0.76249376074704175307859539257,
2.30657838384625996078069108282, 3.88771264048786666034809925860, 4.68755313684334439612235489684, 6.44464104865362991986239792935, 7.75549703156030026352355146874, 8.212333687374287052917123440502, 9.254025542471679652760007981909, 9.818390540351721501623747849826, 11.57787410021481488917627015908, 12.34849049947856212182169534787