L(s) = 1 | + (0.439 − 0.761i)2-s + (0.613 + 1.06i)4-s + (−1.93 − 3.35i)5-s + (1.09 − 1.89i)7-s + 2.83·8-s − 3.41·10-s + (−0.0812 + 0.140i)11-s + (−1.20 − 2.08i)13-s + (−0.960 − 1.66i)14-s + (0.0209 − 0.0362i)16-s + 3·17-s + 3.59·19-s + (2.37 − 4.12i)20-s + (0.0714 + 0.123i)22-s + (−1.41 − 2.45i)23-s + ⋯ |
L(s) = 1 | + (0.310 − 0.538i)2-s + (0.306 + 0.531i)4-s + (−0.867 − 1.50i)5-s + (0.412 − 0.715i)7-s + 1.00·8-s − 1.07·10-s + (−0.0244 + 0.0424i)11-s + (−0.334 − 0.579i)13-s + (−0.256 − 0.444i)14-s + (0.00523 − 0.00906i)16-s + 0.727·17-s + 0.825·19-s + (0.532 − 0.921i)20-s + (0.0152 + 0.0263i)22-s + (−0.295 − 0.512i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11498 - 0.935584i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11498 - 0.935584i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.93 + 3.35i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 1.89i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0812 - 0.140i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.20 + 2.08i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.35 - 5.81i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.59 - 4.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + (-2.90 - 5.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.11 + 5.39i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.69 - 6.40i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 + (-2.56 - 4.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.89 + 3.27i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.93 - 5.07i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 + (-0.634 + 1.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.23 + 7.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.72T + 89T^{2} \) |
| 97 | \( 1 + (-1.95 + 3.38i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14010100021807217156725003483, −11.19367838957481517556784548568, −10.20023069886119179390523825482, −8.805147565833705476116494063703, −7.896874613833659631517435676703, −7.30047233105987494304834424223, −5.23591009169875842507582832343, −4.38081312457330384437325065946, −3.32859874121852764116813844782, −1.24422122659166849697688021546,
2.31587401168300181589833678732, 3.80975636278497073286062357222, 5.30142074578378646748794513024, 6.31836484524024178368009372907, 7.30019113503701205806035151723, 7.931719553792564996207976393848, 9.605438213099162090633668517345, 10.52583720286601080912038384257, 11.52451264923225209680545675864, 11.88429103009151146784136129190