L(s) = 1 | − 0.585·3-s + 9.03i·5-s − 2.64i·7-s − 8.65·9-s − 12.4·11-s − 9.03i·13-s − 5.29i·15-s + 12.3·17-s − 28.8·19-s + 1.54i·21-s + 24.6i·23-s − 56.5·25-s + 10.3·27-s + 22.4i·29-s + 16.7i·31-s + ⋯ |
L(s) = 1 | − 0.195·3-s + 1.80i·5-s − 0.377i·7-s − 0.961·9-s − 1.13·11-s − 0.694i·13-s − 0.352i·15-s + 0.726·17-s − 1.51·19-s + 0.0738i·21-s + 1.07i·23-s − 2.26·25-s + 0.383·27-s + 0.774i·29-s + 0.541i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.110935 + 0.607279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.110935 + 0.607279i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 0.585T + 9T^{2} \) |
| 5 | \( 1 - 9.03iT - 25T^{2} \) |
| 11 | \( 1 + 12.4T + 121T^{2} \) |
| 13 | \( 1 + 9.03iT - 169T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 + 28.8T + 361T^{2} \) |
| 23 | \( 1 - 24.6iT - 529T^{2} \) |
| 29 | \( 1 - 22.4iT - 841T^{2} \) |
| 31 | \( 1 - 16.7iT - 961T^{2} \) |
| 37 | \( 1 - 16.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.97T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.19iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.01iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 30.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 15.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 78.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 17.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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
−12.38613350417916851149909192443, −11.05749074188543220093535889376, −10.76613386397658514879895170958, −9.901230113248688483969989513604, −8.260404861765974515613790328544, −7.41245061528077777686058289450, −6.35926626301545781119199235191, −5.38204713638352848394000394657, −3.47947332509970968791082260630, −2.58025481637514242409427015417,
0.32103442146383376525270727863, 2.27441102717383305559999103030, 4.30917080448361251093015817601, 5.26355230200992111137123162311, 6.12774055583072743141955483198, 7.993663960780479254245842136764, 8.569815113634751333028320854399, 9.433433574077733217975255543543, 10.69509529813213666595694561870, 11.82813430929697822503950407852