L(s) = 1 | + (2.90 + 1.20i)3-s + (1.21 + 2.92i)5-s + (−0.933 + 0.933i)7-s + (4.84 + 4.84i)9-s + (1.05 − 0.435i)11-s + (1.09 − 2.65i)13-s + 9.94i·15-s − 1.61i·17-s + (2.34 − 5.66i)19-s + (−3.82 + 1.58i)21-s + (−1.67 − 1.67i)23-s + (−3.56 + 3.56i)25-s + (4.62 + 11.1i)27-s + (−7.06 − 2.92i)29-s − 1.53·31-s + ⋯ |
L(s) = 1 | + (1.67 + 0.693i)3-s + (0.542 + 1.30i)5-s + (−0.352 + 0.352i)7-s + (1.61 + 1.61i)9-s + (0.317 − 0.131i)11-s + (0.304 − 0.735i)13-s + 2.56i·15-s − 0.390i·17-s + (0.538 − 1.29i)19-s + (−0.835 + 0.346i)21-s + (−0.350 − 0.350i)23-s + (−0.712 + 0.712i)25-s + (0.890 + 2.15i)27-s + (−1.31 − 0.543i)29-s − 0.274·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.083264160\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.083264160\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-2.90 - 1.20i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.21 - 2.92i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (0.933 - 0.933i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.05 + 0.435i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 2.65i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 1.61iT - 17T^{2} \) |
| 19 | \( 1 + (-2.34 + 5.66i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (1.67 + 1.67i)T + 23iT^{2} \) |
| 29 | \( 1 + (7.06 + 2.92i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 + (-2.08 - 5.04i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (1.03 + 1.03i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.98 - 1.64i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 2.97iT - 47T^{2} \) |
| 53 | \( 1 + (10.1 - 4.21i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (3.34 + 8.08i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-12.6 - 5.23i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (11.8 + 4.91i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-5.88 + 5.88i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 + (1.13 - 2.73i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (6.56 - 6.56i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−9.806505824076313292493140707433, −9.481080874497376855701149841051, −8.607419565360185391560583159636, −7.71333165976657129615296501877, −6.94501621192834715726576557970, −5.93938588024219211368972833765, −4.67550127353036450676492577732, −3.40475021155133338530578572965, −2.98204654611243260246573580168, −2.10387504127283683561812703788,
1.36705537474542317708740609145, 1.94652868203560751569436111991, 3.49235576491085591873367530127, 4.10205235913658624354635280169, 5.49174067683302195165645318524, 6.55570853177896422371244993111, 7.50001162916150459938764708659, 8.201120336360590129241721702694, 8.978238732266193240731360148340, 9.424857497172749731982297317884