L(s) = 1 | + i·3-s + (1.96 − 1.06i)5-s − 4.32i·7-s − 9-s + 3.68·11-s + 4.58i·13-s + (1.06 + 1.96i)15-s + 7.89i·17-s + 4.22·19-s + 4.32·21-s + 1.04i·23-s + (2.73 − 4.18i)25-s − i·27-s − 5.93·29-s + 5.66·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.879 − 0.476i)5-s − 1.63i·7-s − 0.333·9-s + 1.11·11-s + 1.27i·13-s + (0.274 + 0.507i)15-s + 1.91i·17-s + 0.969·19-s + 0.943·21-s + 0.217i·23-s + (0.546 − 0.837i)25-s − 0.192i·27-s − 1.10·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.501426300\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.501426300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 4.32iT - 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 7.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 - 1.04iT - 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 1.36T + 41T^{2} \) |
| 43 | \( 1 - 6.90iT - 43T^{2} \) |
| 47 | \( 1 - 0.227iT - 47T^{2} \) |
| 53 | \( 1 + 4.37iT - 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 73 | \( 1 + 1.49iT - 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 + 8.15iT - 97T^{2} \) |
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
−8.597044208224383534118366563821, −7.87219954263977055119454388185, −6.77168343041298546325325420799, −6.44864557175891020496031822188, −5.52007971364372864820088890132, −4.41725981581761549995302656847, −4.15974604820896329885924733797, −3.28947109565025211708774302307, −1.71749510121107121110451758835, −1.16053937245296066648801403159,
0.818494526792188873950793091486, 2.09481271452412841475475105986, 2.68832337236500103396630635852, 3.41714729425995173976535338507, 4.97445864874888956462198503641, 5.67000471941090705057851228402, 5.94326831293843505378002509220, 6.99874358542033789996393132756, 7.45547982046485021128662941936, 8.519158483123005920008402010386