| L(s) = 1 | + 2.51i·3-s + (−1.92 + 1.13i)5-s + 4.64i·7-s − 3.32·9-s − 1.64·11-s + 1.91i·13-s + (−2.85 − 4.84i)15-s − 0.969i·17-s + 4.91·19-s − 11.6·21-s + i·23-s + (2.42 − 4.37i)25-s − 0.825i·27-s − 7.48·29-s + 7.77·31-s + ⋯ |
| L(s) = 1 | + 1.45i·3-s + (−0.861 + 0.507i)5-s + 1.75i·7-s − 1.10·9-s − 0.497·11-s + 0.531i·13-s + (−0.737 − 1.25i)15-s − 0.235i·17-s + 1.12·19-s − 2.54·21-s + 0.208i·23-s + (0.484 − 0.874i)25-s − 0.158i·27-s − 1.38·29-s + 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.262685 - 0.963327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.262685 - 0.963327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.92 - 1.13i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.51iT - 3T^{2} \) |
| 7 | \( 1 - 4.64iT - 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 1.91iT - 13T^{2} \) |
| 17 | \( 1 + 0.969iT - 17T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 7.63iT - 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 3.77iT - 43T^{2} \) |
| 47 | \( 1 - 2.22iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 + 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 3.01T + 71T^{2} \) |
| 73 | \( 1 - 16.6iT - 73T^{2} \) |
| 79 | \( 1 - 0.163T + 79T^{2} \) |
| 83 | \( 1 + 5.66iT - 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 + 1.79iT - 97T^{2} \) |
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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
−10.62796266776863906168365357007, −9.502116793952672448858517689438, −9.211831728925346566069019263463, −8.228973631965595512144476356310, −7.31923285133587253556553309485, −5.96340018216227318098609065702, −5.25417655790549222492889701794, −4.34479445252919380016514958535, −3.33286043343512768789153136371, −2.49859164138832904845787712033,
0.51141566041035652768769298616, 1.36029963424695064067158615917, 3.07284483187391488677347466288, 4.12404491491261624247221275664, 5.17896996607703502244248610190, 6.47074122086352192950352952633, 7.25620668630095066084844725133, 7.81608490445699147539939154773, 8.214675753621177827411808923077, 9.634226859886940610217016300833
