L(s) = 1 | + (1.5 + 2.59i)3-s + (−5.87 − 10.1i)7-s + (−4.5 + 7.79i)9-s + (13.1 − 7.57i)11-s + (−8.87 + 15.3i)13-s + 15.1i·17-s − 11.2·19-s + (17.6 − 30.5i)21-s + (−29.2 − 16.8i)23-s − 27·27-s + (8.23 − 4.75i)29-s + (−28.1 + 48.6i)31-s + (39.3 + 22.7i)33-s − 14·37-s − 53.2·39-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.838 − 1.45i)7-s + (−0.5 + 0.866i)9-s + (1.19 − 0.688i)11-s + (−0.682 + 1.18i)13-s + 0.891i·17-s − 0.592·19-s + (0.838 − 1.45i)21-s + (−1.27 − 0.733i)23-s − 27-s + (0.284 − 0.164i)29-s + (−0.906 + 1.57i)31-s + (1.19 + 0.688i)33-s − 0.378·37-s − 1.36·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2995169177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2995169177\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (5.87 + 10.1i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-13.1 + 7.57i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (8.87 - 15.3i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 15.1iT - 289T^{2} \) |
| 19 | \( 1 + 11.2T + 361T^{2} \) |
| 23 | \( 1 + (29.2 + 16.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-8.23 + 4.75i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (28.1 - 48.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 14T + 1.36e3T^{2} \) |
| 41 | \( 1 + (22.5 + 12.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (9.99 + 17.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (62.6 - 36.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 37.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (55.1 + 31.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (0.618 + 1.07i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-42.9 + 74.4i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 22.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (51.6 + 89.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-78 + 45.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 12.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-49.8 - 86.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.36386525952709362942788249715, −9.523250024678345119311319378315, −8.873304665428675534473582213229, −7.956534653535637095107769986262, −6.80545397228940261556600705744, −6.27671508179489740543241198405, −4.70402126495135318967294809862, −3.94685821346819182096405954922, −3.39889213775023137044361227076, −1.76990253087900819138383100948,
0.083806940619888558118044741468, 1.87326815575720668527019014344, 2.72383767258915603517396887618, 3.72584242313726120986194453006, 5.29299989241751866992758708943, 6.13165309900273483910042483003, 6.86306727801139383849454813930, 7.80979620662333771201867913580, 8.622592249626079315390625653477, 9.547947316370626970785445548085