| L(s) = 1 | + 4.90i·3-s − 6.54·5-s + 5.90i·7-s − 15.0·9-s − 14.3i·11-s − 22.2·13-s − 32.1i·15-s + 28.3·17-s − 4.35i·19-s − 28.9·21-s + 17.8i·23-s + 17.8·25-s − 29.8i·27-s − 32.6·29-s − 35.9i·31-s + ⋯ |
| L(s) = 1 | + 1.63i·3-s − 1.30·5-s + 0.843i·7-s − 1.67·9-s − 1.30i·11-s − 1.71·13-s − 2.14i·15-s + 1.66·17-s − 0.229i·19-s − 1.37·21-s + 0.776i·23-s + 0.715·25-s − 1.10i·27-s − 1.12·29-s − 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6536354740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6536354740\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 3 | \( 1 - 4.90iT - 9T^{2} \) |
| 5 | \( 1 + 6.54T + 25T^{2} \) |
| 7 | \( 1 - 5.90iT - 49T^{2} \) |
| 11 | \( 1 + 14.3iT - 121T^{2} \) |
| 13 | \( 1 + 22.2T + 169T^{2} \) |
| 17 | \( 1 - 28.3T + 289T^{2} \) |
| 23 | \( 1 - 17.8iT - 529T^{2} \) |
| 29 | \( 1 + 32.6T + 841T^{2} \) |
| 31 | \( 1 + 35.9iT - 961T^{2} \) |
| 37 | \( 1 - 52.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 58.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 64.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.97T + 3.72e3T^{2} \) |
| 67 | \( 1 - 64.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 7.10iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 12.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 50.7T + 9.40e3T^{2} \) |
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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
−9.582124703606939723172401278123, −8.869078671275850621699825910440, −8.023087801550178835173563621474, −7.35751427413472047307730837256, −5.63959716083848308010074089308, −5.35253265808244516364138989651, −4.20180254525933913423169620333, −3.52080939016568960211237815862, −2.73083028409959710150061304102, −0.26603313491136558676841058547,
0.829236778416248688853708372867, 2.01939273951380935096299801154, 3.20627014655166067505103769226, 4.35316671116326107337281997744, 5.28434394592819278582721084643, 6.72640755896186084878480206386, 7.22286425295620315644475251035, 7.72353522860403320327184561650, 8.154862696229350266626442269702, 9.615962081312440924414619529993
