L(s) = 1 | + (−2.10 + 1.76i)3-s + (−0.939 − 0.342i)5-s + (−1.23 + 2.14i)7-s + (0.790 − 4.48i)9-s + (0.186 + 0.322i)11-s + (−2.64 − 2.21i)13-s + (2.58 − 0.940i)15-s + (−0.361 − 2.05i)17-s + (0.0754 − 4.35i)19-s + (−1.18 − 6.70i)21-s + (−0.815 + 0.296i)23-s + (0.766 + 0.642i)25-s + (2.13 + 3.70i)27-s + (1.80 − 10.2i)29-s + (−0.935 + 1.62i)31-s + ⋯ |
L(s) = 1 | + (−1.21 + 1.02i)3-s + (−0.420 − 0.152i)5-s + (−0.467 + 0.810i)7-s + (0.263 − 1.49i)9-s + (0.0562 + 0.0973i)11-s + (−0.732 − 0.614i)13-s + (0.666 − 0.242i)15-s + (−0.0877 − 0.497i)17-s + (0.0173 − 0.999i)19-s + (−0.257 − 1.46i)21-s + (−0.170 + 0.0618i)23-s + (0.153 + 0.128i)25-s + (0.411 + 0.712i)27-s + (0.334 − 1.89i)29-s + (−0.168 + 0.291i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0674304 - 0.0964561i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0674304 - 0.0964561i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.0754 + 4.35i)T \) |
good | 3 | \( 1 + (2.10 - 1.76i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (1.23 - 2.14i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.186 - 0.322i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.64 + 2.21i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.361 + 2.05i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.815 - 0.296i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.80 + 10.2i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.935 - 1.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 + (6.08 - 5.10i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.27 + 1.19i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.871 - 4.94i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.54 + 2.74i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.44 + 8.18i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (12.1 - 4.42i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.28 - 7.31i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.31 + 0.844i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.87 - 3.25i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (11.9 - 9.99i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.14 - 5.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.811 + 0.680i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.49 + 14.1i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.19169136303932865733077832233, −10.10996234607237058175458418791, −9.564795362444431450771634334394, −8.475955394345835371370776595724, −7.10473089493297885404679335188, −6.00139398000516539654687721414, −5.15661000643997072519573257096, −4.34302971963499136339591761650, −2.88398827360393141804576004386, −0.090784322932838528747600465566,
1.60302020002330042788057809140, 3.60260701654841659801020816990, 4.94075204215777307160653495272, 6.08529514881699287580334887424, 6.97663433996894240719918894879, 7.43810163236729405529769045596, 8.740514353979879820067374141258, 10.24230584052478991162112822270, 10.73218562282358994688532212496, 11.93269448338930411646374803688