L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.05 + 0.889i)5-s + (0.970 − 2.46i)7-s − 0.999·8-s + (1.79 − 1.33i)10-s − 2.28i·11-s + (2.02 − 3.49i)13-s + (−1.64 − 2.07i)14-s + (−0.5 + 0.866i)16-s + (−5.58 − 3.22i)17-s + (−0.177 + 0.102i)19-s + (−0.255 − 2.22i)20-s + (−1.98 − 1.14i)22-s − 2.00·23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.917 + 0.397i)5-s + (0.366 − 0.930i)7-s − 0.353·8-s + (0.567 − 0.421i)10-s − 0.689i·11-s + (0.560 − 0.970i)13-s + (−0.439 − 0.553i)14-s + (−0.125 + 0.216i)16-s + (−1.35 − 0.782i)17-s + (−0.0407 + 0.0235i)19-s + (−0.0572 − 0.496i)20-s + (−0.422 − 0.243i)22-s − 0.418·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.217683275\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.217683275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.05 - 0.889i)T \) |
| 7 | \( 1 + (-0.970 + 2.46i)T \) |
good | 11 | \( 1 + 2.28iT - 11T^{2} \) |
| 13 | \( 1 + (-2.02 + 3.49i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (5.58 + 3.22i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.177 - 0.102i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 + (2.95 - 1.70i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.844 - 0.487i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.40 + 4.27i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.12 - 1.22i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.13 + 2.38i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.35 + 4.08i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.53 - 7.85i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.33 - 5.39i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.24 - 3.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 + (-7.78 + 13.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.61 + 6.26i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.91 - 5.72i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.08 - 8.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.24 + 7.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.106953830567870120080570406822, −8.275016358147726109978328395169, −7.26996327892094416781145577991, −6.42532486260136752581099983952, −5.66624388100748107882898100963, −4.82942299366562545848330272187, −3.82411611724576511721409041627, −2.94633909342826653354997487012, −1.93193660672603627770485381003, −0.69960487337955385312250281207,
1.74031600027041299360093238019, 2.42582282296280102068705338507, 4.01419814131871632741910453619, 4.70970328959369117755683848694, 5.55998227699694814049353758167, 6.30306416807491780671560002992, 6.82647891746804212678219728455, 8.093713483001010644835139571087, 8.672095902039021556896793716350, 9.317555155700915865925024007842