| L(s) = 1 | + (0.324 − 3.71i)2-s + (−9.72 − 1.71i)4-s + (4.74 + 1.56i)5-s + (7.47 + 5.23i)7-s + (−5.65 + 21.1i)8-s + (7.35 − 17.1i)10-s + (7.46 + 2.71i)11-s + (−0.449 − 5.13i)13-s + (21.8 − 26.0i)14-s + (39.4 + 14.3i)16-s + (−7.59 − 28.3i)17-s + (9.25 − 5.34i)19-s + (−43.4 − 23.3i)20-s + (12.5 − 26.8i)22-s + (23.0 − 16.1i)23-s + ⋯ |
| L(s) = 1 | + (0.162 − 1.85i)2-s + (−2.43 − 0.428i)4-s + (0.949 + 0.313i)5-s + (1.06 + 0.747i)7-s + (−0.707 + 2.64i)8-s + (0.735 − 1.71i)10-s + (0.678 + 0.247i)11-s + (−0.0345 − 0.394i)13-s + (1.55 − 1.85i)14-s + (2.46 + 0.896i)16-s + (−0.446 − 1.66i)17-s + (0.487 − 0.281i)19-s + (−2.17 − 1.16i)20-s + (0.568 − 1.21i)22-s + (1.00 − 0.700i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.944272 - 1.93326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.944272 - 1.93326i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-4.74 - 1.56i)T \) |
| good | 2 | \( 1 + (-0.324 + 3.71i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-7.47 - 5.23i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-7.46 - 2.71i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.449 + 5.13i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (7.59 + 28.3i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-9.25 + 5.34i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-23.0 + 16.1i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-19.8 - 23.6i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (3.45 - 19.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-6.34 - 23.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (33.8 + 28.3i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (9.22 + 19.7i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-26.5 - 18.5i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (36.3 + 36.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (13.3 + 36.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 6.31i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-53.2 + 4.65i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-20.4 + 35.3i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (14.7 - 55.2i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-32.7 - 38.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (121. + 10.6i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (77.2 - 44.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-104. + 48.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−10.91583626698220063162959385559, −9.987574229662893799986329359221, −9.170869737390422591094700900745, −8.602170472594776434428771401089, −6.88150690725251599573481447884, −5.20889346777481613674017961859, −4.84618650914143331495425467700, −3.12197223408800395057433375302, −2.29014376800441541263104503344, −1.16248150495140434131695498127,
1.35148365594659363395290864949, 3.99629080919547981402142521763, 4.80658312543908130635536784626, 5.83236019590397255937130263939, 6.54383397617364495570400963002, 7.57177939449236060478672352250, 8.408106966239862470510088602433, 9.140216495546309135634595891954, 10.11493404635749580029837529075, 11.31199080458742333525072333217
