| L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.23i·5-s + 2.44i·6-s − 2.82·8-s − 2.99·9-s − 3.16i·10-s − 10.8·11-s − 3.46i·12-s + 19.2i·13-s + 3.87·15-s + 4.00·16-s − 10.2i·17-s + 4.24·18-s + 20.8i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.447i·5-s + 0.408i·6-s − 0.353·8-s − 0.333·9-s − 0.316i·10-s − 0.983·11-s − 0.288i·12-s + 1.48i·13-s + 0.258·15-s + 0.250·16-s − 0.604i·17-s + 0.235·18-s + 1.09i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3841782723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3841782723\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 10.8T + 121T^{2} \) |
| 13 | \( 1 - 19.2iT - 169T^{2} \) |
| 17 | \( 1 + 10.2iT - 289T^{2} \) |
| 19 | \( 1 - 20.8iT - 361T^{2} \) |
| 23 | \( 1 - 21.0T + 529T^{2} \) |
| 29 | \( 1 - 19.0T + 841T^{2} \) |
| 31 | \( 1 + 40.0iT - 961T^{2} \) |
| 37 | \( 1 + 50.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.95T + 2.80e3T^{2} \) |
| 59 | \( 1 + 28.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 70.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 19.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 49.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 132. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 90.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 39.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.6iT - 9.40e3T^{2} \) |
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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
−8.899252095526060086539771191816, −8.234319475938549364792729279484, −7.31011019223096436097037824132, −6.87379008207673811867958619392, −5.94992833889444901373697870439, −4.95385577606942337971203174124, −3.61023459983507127764763922256, −2.50321070872422726311787980075, −1.66414883627657576353897619211, −0.14849775018869043542715958125,
1.07923141999146238282182539292, 2.64582412723922556118417652926, 3.38022018485052602994778146977, 4.88836190905768715112498496143, 5.29721606151216339977887893081, 6.43600932733019281895964802501, 7.42385730640928026395482171802, 8.297906682708991509136361969624, 8.706943779688082090843296592786, 9.659288861338988774536515911307
