| L(s) = 1 | + (−5.20 + 1.39i)2-s + (2.56 + 0.687i)3-s + (11.2 − 6.49i)4-s + (−19.5 + 15.5i)5-s − 14.2·6-s + (20.1 − 44.6i)7-s + (11.4 − 11.4i)8-s + (−64.0 − 36.9i)9-s + (80.2 − 108. i)10-s + (−91.3 − 158. i)11-s + (33.3 − 8.92i)12-s + (141. − 141. i)13-s + (−42.6 + 260. i)14-s + (−60.9 + 26.3i)15-s + (−147. + 255. i)16-s + (−110. + 412. i)17-s + ⋯ |
| L(s) = 1 | + (−1.30 + 0.348i)2-s + (0.285 + 0.0763i)3-s + (0.702 − 0.405i)4-s + (−0.783 + 0.621i)5-s − 0.397·6-s + (0.411 − 0.911i)7-s + (0.179 − 0.179i)8-s + (−0.790 − 0.456i)9-s + (0.802 − 1.08i)10-s + (−0.755 − 1.30i)11-s + (0.231 − 0.0619i)12-s + (0.839 − 0.839i)13-s + (−0.217 + 1.32i)14-s + (−0.270 + 0.117i)15-s + (−0.576 + 0.998i)16-s + (−0.382 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.226401 - 0.272470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.226401 - 0.272470i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (19.5 - 15.5i)T \) |
| 7 | \( 1 + (-20.1 + 44.6i)T \) |
| good | 2 | \( 1 + (5.20 - 1.39i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-2.56 - 0.687i)T + (70.1 + 40.5i)T^{2} \) |
| 11 | \( 1 + (91.3 + 158. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-141. + 141. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (110. - 412. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (213. + 123. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (132. + 492. i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 - 1.08e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (227. + 393. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-72.2 + 19.3i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + 1.37e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (493. - 493. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-826. + 221. i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (2.89e3 + 775. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (-1.51e3 + 873. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.63e3 + 2.84e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (700. - 2.61e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 - 242.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-4.72e3 - 1.26e3i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (-3.16e3 - 1.82e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-7.24e3 + 7.24e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-2.12e3 - 1.22e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.79e3 - 1.79e3i)T + 8.85e7iT^{2} \) |
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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
−15.70751857724884116951567815080, −14.58059866019350076713167495135, −13.17130055638129800516957212127, −10.95610349710910306740026277286, −10.61360422852097779079415601721, −8.476339008813339529809912613102, −8.119870349592942288694078582528, −6.46132339100086679773400240655, −3.60560035586883161628956512788, −0.35938655289280206779674362983,
2.09721181712378150057769569459, 4.95148914765805576771904520466, 7.61691808461394555347526521459, 8.547319602452921352499937602837, 9.468618926703003704112924149363, 11.17855798786818095924922727710, 11.98432727616483103840005910850, 13.68639921973704444395696582066, 15.28244582084376274744112163486, 16.28687745419114477887954685702
