L(s) = 1 | − 1.70i·2-s − 3i·3-s + 5.10·4-s + (−8.10 + 7.70i)5-s − 5.10·6-s + 22.2i·7-s − 22.2i·8-s − 9·9-s + (13.1 + 13.7i)10-s − 1.79·11-s − 15.3i·12-s − 58.2i·13-s + 37.7·14-s + (23.1 + 24.3i)15-s + 2.89·16-s + 18.9i·17-s + ⋯ |
L(s) = 1 | − 0.601i·2-s − 0.577i·3-s + 0.638·4-s + (−0.724 + 0.688i)5-s − 0.347·6-s + 1.19i·7-s − 0.985i·8-s − 0.333·9-s + (0.414 + 0.436i)10-s − 0.0490·11-s − 0.368i·12-s − 1.24i·13-s + 0.721·14-s + (0.397 + 0.418i)15-s + 0.0452·16-s + 0.270i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.724i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.688 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.940177 - 0.403552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.940177 - 0.403552i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (8.10 - 7.70i)T \) |
good | 2 | \( 1 + 1.70iT - 8T^{2} \) |
| 7 | \( 1 - 22.2iT - 343T^{2} \) |
| 11 | \( 1 + 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.8iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 384. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 173. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 384. iT - 9.12e5T^{2} \) |
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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
−19.05854589981619490520107685956, −17.84825338303029670933596598148, −15.78216246636905466739539400184, −14.92608534778043455477058147175, −12.72268697022107761520899556525, −11.79519540073729265522515856540, −10.51178912745310602829285786506, −8.125903885799567260777787983999, −6.37652257153146061052060194443, −2.79345628644283790574684224363,
4.43486437003440839041916273830, 6.86705334043097671359123212326, 8.476519061928771228668739641514, 10.58724244716950306665003513674, 11.93352692971075732567821740168, 13.97056013694459173859190634953, 15.37221282991038364890764394428, 16.48025858588084541219955233814, 17.05042483898276986362979849657, 19.42381086195133821834902176725