L(s) = 1 | + 2.35i·2-s − 3.55·4-s + 3.69i·5-s − 0.801i·7-s − 3.66i·8-s − 8.70·10-s + 2.85i·11-s + 1.89·14-s + 1.52·16-s + 2.93·17-s + 2.44i·19-s − 13.1i·20-s − 6.71·22-s − 7.78·23-s − 8.63·25-s + ⋯ |
L(s) = 1 | + 1.66i·2-s − 1.77·4-s + 1.65i·5-s − 0.303i·7-s − 1.29i·8-s − 2.75·10-s + 0.859i·11-s + 0.505·14-s + 0.381·16-s + 0.712·17-s + 0.560i·19-s − 2.93i·20-s − 1.43·22-s − 1.62·23-s − 1.72·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9359201103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9359201103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.35iT - 2T^{2} \) |
| 5 | \( 1 - 3.69iT - 5T^{2} \) |
| 7 | \( 1 + 0.801iT - 7T^{2} \) |
| 11 | \( 1 - 2.85iT - 11T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 + 2.34iT - 31T^{2} \) |
| 37 | \( 1 - 7.44iT - 37T^{2} \) |
| 41 | \( 1 + 0.850iT - 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 2.44iT - 47T^{2} \) |
| 53 | \( 1 - 9.96T + 53T^{2} \) |
| 59 | \( 1 - 5.38iT - 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 - 8.12iT - 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 7.04iT - 83T^{2} \) |
| 89 | \( 1 + 1.13iT - 89T^{2} \) |
| 97 | \( 1 - 5.94iT - 97T^{2} \) |
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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
−10.05651932795061784639571952774, −9.189833052945895034285577700176, −7.909316964484153528692709518387, −7.64303699838596872026951589921, −6.88879179985235424258210185718, −6.22550411743112477934166812174, −5.57819231590894927796438893392, −4.36080469016562574588959453430, −3.53081691537283655566765211294, −2.17104215548601367344215788341,
0.38219061254588575890010683466, 1.37316367862143324126032901249, 2.40493117856235023151989027897, 3.63363874953090742454773412152, 4.30613812988014028224377894392, 5.27389846330160119624110439135, 5.92406443172446175622632301933, 7.60292297205667884567578696711, 8.558493824372101891413849296708, 8.966864587670790561553631526170