| L(s) = 1 | + (−1.35 + 0.784i)3-s + (38.3 + 22.1i)5-s + (42.3 − 24.5i)7-s + (−39.2 + 68.0i)9-s + (−11.7 − 20.3i)11-s − 136. i·13-s − 69.4·15-s + (−227. + 131. i)17-s + (−387. − 223. i)19-s + (−38.3 + 66.6i)21-s + (374. − 648. i)23-s + (667. + 1.15e3i)25-s − 250. i·27-s + 406.·29-s + (−584. + 337. i)31-s + ⋯ |
| L(s) = 1 | + (−0.151 + 0.0871i)3-s + (1.53 + 0.885i)5-s + (0.865 − 0.501i)7-s + (−0.484 + 0.839i)9-s + (−0.0971 − 0.168i)11-s − 0.806i·13-s − 0.308·15-s + (−0.786 + 0.454i)17-s + (−1.07 − 0.619i)19-s + (−0.0869 + 0.151i)21-s + (0.708 − 1.22i)23-s + (1.06 + 1.84i)25-s − 0.343i·27-s + 0.483·29-s + (−0.607 + 0.350i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.55340 + 0.313612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55340 + 0.313612i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-42.3 + 24.5i)T \) |
| good | 3 | \( 1 + (1.35 - 0.784i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-38.3 - 22.1i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (11.7 + 20.3i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + 136. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (227. - 131. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (387. + 223. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-374. + 648. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 406.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (584. - 337. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (372. - 645. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.47e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.63e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (579. + 334. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-1.01e3 - 1.75e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.01e3 - 585. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.02e3 + 1.16e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.77e3 - 6.54e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-3.28e3 + 1.89e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.89e3 - 6.75e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.18e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-8.05e3 - 4.64e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 9.98e3iT - 8.85e7T^{2} \) |
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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
−16.99277902597652870338454631373, −15.03647174950760246300832044207, −14.04691261900454492100049185574, −13.14126412476617304548406991666, −10.85936506646538165828397582482, −10.49092140934562258115353080819, −8.540182932206538752322731057512, −6.67792386087337301726429095801, −5.16750799135441965390962438282, −2.31532916365648205235693675108,
1.80689940933409746069633287096, 5.00667049790379191886197397047, 6.30597490567660162390187619599, 8.674760391305448618824857250066, 9.558582809921076026651889991812, 11.38606255032621461100265094830, 12.68642173916509162703112111361, 13.86697844049336044157045620747, 15.00292285037885964820613862467, 16.74446766843733654455122525384
