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
−9.659288861338988774536515911307, −8.706943779688082090843296592786, −8.297906682708991509136361969624, −7.42385730640928026395482171802, −6.43600932733019281895964802501, −5.29721606151216339977887893081, −4.88836190905768715112498496143, −3.38022018485052602994778146977, −2.64582412723922556118417652926, −1.07923141999146238282182539292,
0.14849775018869043542715958125, 1.66414883627657576353897619211, 2.50321070872422726311787980075, 3.61023459983507127764763922256, 4.95385577606942337971203174124, 5.94992833889444901373697870439, 6.87379008207673811867958619392, 7.31011019223096436097037824132, 8.234319475938549364792729279484, 8.899252095526060086539771191816