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
−11.93269448338930411646374803688, −10.73218562282358994688532212496, −10.24230584052478991162112822270, −8.740514353979879820067374141258, −7.43810163236729405529769045596, −6.97663433996894240719918894879, −6.08529514881699287580334887424, −4.94075204215777307160653495272, −3.60260701654841659801020816990, −1.60302020002330042788057809140,
0.090784322932838528747600465566, 2.88398827360393141804576004386, 4.34302971963499136339591761650, 5.15661000643997072519573257096, 6.00139398000516539654687721414, 7.10473089493297885404679335188, 8.475955394345835371370776595724, 9.564795362444431450771634334394, 10.10996234607237058175458418791, 11.19169136303932865733077832233