L(s) = 1 | + (2.37 − 1.83i)3-s + (7.15 − 4.13i)7-s + (2.23 − 8.71i)9-s + (−1.19 + 0.687i)11-s + (−18.6 − 10.7i)13-s − 25.3·17-s + 2.49·19-s + (9.36 − 22.9i)21-s + (19.1 − 33.1i)23-s + (−10.7 − 24.7i)27-s + (−37.0 + 21.4i)29-s + (10.9 − 18.9i)31-s + (−1.55 + 3.81i)33-s + 30.5i·37-s + (−64.0 + 8.78i)39-s + ⋯ |
L(s) = 1 | + (0.790 − 0.612i)3-s + (1.02 − 0.590i)7-s + (0.248 − 0.968i)9-s + (−0.108 + 0.0624i)11-s + (−1.43 − 0.829i)13-s − 1.48·17-s + 0.131·19-s + (0.445 − 1.09i)21-s + (0.833 − 1.44i)23-s + (−0.397 − 0.917i)27-s + (−1.27 + 0.738i)29-s + (0.353 − 0.611i)31-s + (−0.0472 + 0.115i)33-s + 0.826i·37-s + (−1.64 + 0.225i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.083057977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083057977\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.37 + 1.83i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-7.15 + 4.13i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (1.19 - 0.687i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (18.6 + 10.7i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 25.3T + 289T^{2} \) |
| 19 | \( 1 - 2.49T + 361T^{2} \) |
| 23 | \( 1 + (-19.1 + 33.1i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (37.0 - 21.4i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-10.9 + 18.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 30.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-7.29 - 4.21i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-61.6 + 35.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-24.3 - 42.1i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 1.96T + 2.80e3T^{2} \) |
| 59 | \( 1 + (3.77 + 2.18i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (18.4 + 32.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.9 + 8.06i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 71.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (3.98 + 6.90i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-52.1 - 90.2i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 9.37iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-11.8 + 6.86i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.420643174450855770476564080192, −8.721268711839506106543785280597, −7.75086495108208754201960549313, −7.36337917515887959519279586360, −6.39169663857694501022046514530, −4.99506630450933089083338069311, −4.26908953146848858558220204955, −2.85452731393091043098160083172, −2.00138703567392008653973444391, −0.57123692541851096928503499176,
1.89714110066784997560732256261, 2.59594398624591959584944825363, 4.03982528134842245892328536858, 4.79815583030862780793674818219, 5.57320147729239636405475855234, 7.14011131398265549585996966368, 7.67801275625802422305509451227, 8.808907680528923468190279660672, 9.183895767011742823382655888814, 10.02377339210254582319804541019