| L(s) = 1 | + 1.73i·3-s − 2.23·5-s + 2.64i·7-s − 2.99·9-s − 12.5i·11-s + 4.20·13-s − 3.87i·15-s + 2.52·17-s − 18.7i·19-s − 4.58·21-s + 36.9i·23-s + 5.00·25-s − 5.19i·27-s + 34.6·29-s + 48.4i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447·5-s + 0.377i·7-s − 0.333·9-s − 1.14i·11-s + 0.323·13-s − 0.258i·15-s + 0.148·17-s − 0.986i·19-s − 0.218·21-s + 1.60i·23-s + 0.200·25-s − 0.192i·27-s + 1.19·29-s + 1.56i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4461393755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4461393755\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 12.5iT - 121T^{2} \) |
| 13 | \( 1 - 4.20T + 169T^{2} \) |
| 17 | \( 1 - 2.52T + 289T^{2} \) |
| 19 | \( 1 + 18.7iT - 361T^{2} \) |
| 23 | \( 1 - 36.9iT - 529T^{2} \) |
| 29 | \( 1 - 34.6T + 841T^{2} \) |
| 31 | \( 1 - 48.4iT - 961T^{2} \) |
| 37 | \( 1 + 44.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 68.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 33.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 28.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 93.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 22.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 91.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 99.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 65.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 172.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 89.1T + 9.40e3T^{2} \) |
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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
−9.489463188624473627581064368804, −8.666866734616338152875569748888, −8.287468433541873106609932714321, −7.15201966264207530298189821618, −6.30407777702140530216206733771, −5.36062838116315832017993626181, −4.70968840081580774733969794214, −3.43788108400488619008755390273, −3.04968917270557690334870536308, −1.35977832912781352916527189529,
0.12294122200370471882611902910, 1.45795701258501972795394711854, 2.51233721291519720040574067227, 3.74016534774083823591799811625, 4.52306964053972733882564721442, 5.52336895823703351236551661032, 6.68111095822090114854460550267, 7.00448762993698465511821518062, 8.157384100931618783372450946665, 8.381387757772624387633839413569
