L(s) = 1 | + (−0.587 + 0.809i)3-s + (−0.900 + 2.04i)5-s − 0.957i·7-s + (−0.309 − 0.951i)9-s + (−1.67 + 5.15i)11-s + (−1.92 + 0.625i)13-s + (−1.12 − 1.93i)15-s + (0.377 + 0.520i)17-s + (−4.07 + 2.96i)19-s + (0.774 + 0.562i)21-s + (−3.34 − 1.08i)23-s + (−3.37 − 3.68i)25-s + (0.951 + 0.309i)27-s + (8.20 + 5.96i)29-s + (−2.98 + 2.16i)31-s + ⋯ |
L(s) = 1 | + (−0.339 + 0.467i)3-s + (−0.402 + 0.915i)5-s − 0.361i·7-s + (−0.103 − 0.317i)9-s + (−0.504 + 1.55i)11-s + (−0.533 + 0.173i)13-s + (−0.290 − 0.498i)15-s + (0.0916 + 0.126i)17-s + (−0.935 + 0.679i)19-s + (0.169 + 0.122i)21-s + (−0.697 − 0.226i)23-s + (−0.675 − 0.737i)25-s + (0.183 + 0.0594i)27-s + (1.52 + 1.10i)29-s + (−0.536 + 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.338879 + 0.709145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.338879 + 0.709145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (0.900 - 2.04i)T \) |
good | 7 | \( 1 + 0.957iT - 7T^{2} \) |
| 11 | \( 1 + (1.67 - 5.15i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.92 - 0.625i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.377 - 0.520i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.07 - 2.96i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.34 + 1.08i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-8.20 - 5.96i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.98 - 2.16i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-10.7 + 3.49i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 3.35i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.766iT - 43T^{2} \) |
| 47 | \( 1 + (-2.90 + 3.99i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.49 - 4.81i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.45 + 4.48i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 4.13i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (5.59 + 7.70i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-9.66 - 7.02i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.16 - 1.67i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.58 - 6.96i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.819 + 1.12i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.527 + 1.62i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 11.8i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.14202338231551341712992086148, −10.89304590375103446551312084428, −10.33741308484352694646561758620, −9.590571247977398543943858909091, −8.088982802163805373248708580852, −7.19520571699066818039703331949, −6.30720905155842263699639229031, −4.82535887387697480424173515653, −3.93063295569729801889838101977, −2.39034550201853076812092023027,
0.58968769592860991218066089144, 2.61483693419786015736300993323, 4.30393985439625756257078226574, 5.46417051336255264063580714667, 6.29786208380862812435706545304, 7.80478272607340599388118367294, 8.366014144461551696671790841490, 9.379285565540139622501540319959, 10.67721135312075201056211933761, 11.57198191418209031309997430218