L(s) = 1 | + 3.20i·2-s + (0.930 − 5.11i)3-s − 2.24·4-s − 11.9·5-s + (16.3 + 2.97i)6-s + 18.4i·8-s + (−25.2 − 9.50i)9-s − 38.3i·10-s + 60.3i·11-s + (−2.08 + 11.4i)12-s + 3.46i·13-s + (−11.1 + 61.2i)15-s − 76.9·16-s − 32.4·17-s + (30.4 − 80.8i)18-s + 142. i·19-s + ⋯ |
L(s) = 1 | + 1.13i·2-s + (0.178 − 0.983i)3-s − 0.280·4-s − 1.07·5-s + (1.11 + 0.202i)6-s + 0.814i·8-s + (−0.935 − 0.352i)9-s − 1.21i·10-s + 1.65i·11-s + (−0.0501 + 0.275i)12-s + 0.0738i·13-s + (−0.191 + 1.05i)15-s − 1.20·16-s − 0.463·17-s + (0.398 − 1.05i)18-s + 1.72i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.147976 + 0.869434i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.147976 + 0.869434i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.930 + 5.11i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 3.20iT - 8T^{2} \) |
| 5 | \( 1 + 11.9T + 125T^{2} \) |
| 11 | \( 1 - 60.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 3.46iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 32.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 89.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 247. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 207. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 98.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 59.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 232.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 465.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.1iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 550. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 372. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 879.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 21.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 612. iT - 9.12e5T^{2} \) |
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
−13.09966381744388585105316518965, −12.02572133179818000495830569786, −11.46352321315988746676937331206, −9.697740146807216441238979515557, −8.152459474168257227351602537244, −7.71361705229995557849100955118, −6.88037650391134527874300391351, −5.75755678810671030168048145061, −4.14345117405642074742479104618, −2.08100094432720455692244011374,
0.40144609638449387156449934666, 2.88787722312911889750921243233, 3.66646044634294304386046854137, 4.91164681129878596715290607700, 6.73031569512668293329857805061, 8.391167214907803738212828274892, 9.117019384484547315727598196488, 10.53301212947781110347894804253, 11.11809650518521882154936214863, 11.67153778968934932255727002020