| L(s) = 1 | − 5.85·3-s + 5.78i·5-s + 2.64i·7-s + 25.2·9-s − 3.01·11-s + 9.78i·13-s − 33.8i·15-s − 11.6·17-s − 25.5·19-s − 15.4i·21-s + 26.1i·23-s − 8.42·25-s − 95.4·27-s + 1.56i·29-s + 12.0i·31-s + ⋯ |
| L(s) = 1 | − 1.95·3-s + 1.15i·5-s + 0.377i·7-s + 2.81·9-s − 0.274·11-s + 0.752i·13-s − 2.25i·15-s − 0.682·17-s − 1.34·19-s − 0.737i·21-s + 1.13i·23-s − 0.337·25-s − 3.53·27-s + 0.0539i·29-s + 0.387i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4495912157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4495912157\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 + 5.85T + 9T^{2} \) |
| 5 | \( 1 - 5.78iT - 25T^{2} \) |
| 11 | \( 1 + 3.01T + 121T^{2} \) |
| 13 | \( 1 - 9.78iT - 169T^{2} \) |
| 17 | \( 1 + 11.6T + 289T^{2} \) |
| 19 | \( 1 + 25.5T + 361T^{2} \) |
| 23 | \( 1 - 26.1iT - 529T^{2} \) |
| 29 | \( 1 - 1.56iT - 841T^{2} \) |
| 31 | \( 1 - 12.0iT - 961T^{2} \) |
| 37 | \( 1 - 70.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 49.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 73.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 44.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 12.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 35.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 24.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 11.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 7.75T + 9.40e3T^{2} \) |
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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
−9.890263536842952362278228821078, −8.988238849240009967483275541436, −7.60765994465299814121949339881, −6.95215872268198942050990687937, −6.27907639287371939131251103925, −5.84565881641554410219862121711, −4.73728895682465157544980065626, −4.09878091634070964905106447711, −2.62021028368257474674988616321, −1.39415136346490970846693421838,
0.24218018467442891866925358245, 0.72985421448597894204766466555, 2.08030103286817084087282556804, 4.19594507722497214442150383625, 4.48090047401733547320135295289, 5.44334549026472600274868586458, 5.97688403079832994295569052200, 6.82474614598968629572066347397, 7.65817641603778901128935482929, 8.644874648018440272926639333576
