| L(s) = 1 | + 3.44i·3-s + 4.88·5-s + 2.64i·7-s − 2.84·9-s − 21.4i·11-s − 13.0·13-s + 16.8i·15-s − 0.234·17-s − 4.55i·19-s − 9.10·21-s − 10.9i·23-s − 1.15·25-s + 21.1i·27-s + 34.6·29-s − 34.1i·31-s + ⋯ |
| L(s) = 1 | + 1.14i·3-s + 0.976·5-s + 0.377i·7-s − 0.315·9-s − 1.95i·11-s − 1.00·13-s + 1.12i·15-s − 0.0138·17-s − 0.239i·19-s − 0.433·21-s − 0.476i·23-s − 0.0463·25-s + 0.784i·27-s + 1.19·29-s − 1.10i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.289340846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289340846\) |
| \(L(2)\) |
|
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 | \( 1 - 3.44iT - 9T^{2} \) |
| 5 | \( 1 - 4.88T + 25T^{2} \) |
| 11 | \( 1 + 21.4iT - 121T^{2} \) |
| 13 | \( 1 + 13.0T + 169T^{2} \) |
| 17 | \( 1 + 0.234T + 289T^{2} \) |
| 19 | \( 1 + 4.55iT - 361T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 - 34.6T + 841T^{2} \) |
| 31 | \( 1 + 34.1iT - 961T^{2} \) |
| 37 | \( 1 - 54.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.84iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 18.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 47.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 159.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 90.4T + 9.40e3T^{2} \) |
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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
−9.160760147045618741232351919303, −8.636012655008531990113329428716, −7.64538052592015696532062589492, −6.36738074631789895553244949096, −5.80245193807716148764651131620, −5.05309130014347080397605238526, −4.17852516205035293849721848715, −3.11169240122702222923941202055, −2.31015453755262041807383502757, −0.64697284539292249597475507311,
1.12201095996524681783516575140, 1.97452111142445493853105918580, 2.64782231137416970799562465913, 4.32210804789445993051830094742, 5.00841001083311635855454172065, 6.13929553792945925651914782078, 6.79622109356589794825852888382, 7.45017950994854967748546550573, 7.963738131547945360436552449557, 9.320660264290006234374791712312
