| L(s) = 1 | − 1.41i·2-s + 1.73·3-s − 2.00·4-s − 2.44i·6-s + (1.46 + 6.84i)7-s + 2.82i·8-s + 2.99·9-s − 5.87·11-s − 3.46·12-s − 2.06·13-s + (9.68 − 2.07i)14-s + 4.00·16-s + 2.59·17-s − 4.24i·18-s − 14.3i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.209 + 0.977i)7-s + 0.353i·8-s + 0.333·9-s − 0.534·11-s − 0.288·12-s − 0.158·13-s + (0.691 − 0.148i)14-s + 0.250·16-s + 0.152·17-s − 0.235i·18-s − 0.757i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.824393372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824393372\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.46 - 6.84i)T \) |
| good | 11 | \( 1 + 5.87T + 121T^{2} \) |
| 13 | \( 1 + 2.06T + 169T^{2} \) |
| 17 | \( 1 - 2.59T + 289T^{2} \) |
| 19 | \( 1 + 14.3iT - 361T^{2} \) |
| 23 | \( 1 - 4.73iT - 529T^{2} \) |
| 29 | \( 1 - 16.1T + 841T^{2} \) |
| 31 | \( 1 - 48.3iT - 961T^{2} \) |
| 37 | \( 1 - 46.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 74.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 39.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 60.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 73.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 39.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 11.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 41.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 92.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.26T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.848606483063954637272194537124, −8.928718180636490849744799775308, −8.453252750763768891617388978739, −7.53526583950942173427849582751, −6.39544221275888769626608660662, −5.23684399162243603843840738544, −4.54948540805332796136683758633, −3.13968619884288077048465584862, −2.57788399289235944816092262667, −1.34167343732068762753815486014,
0.54618151237925699747320891498, 2.13291411240515335592146659980, 3.56850700047125847458461247760, 4.30915606467987035870038885107, 5.35124557327359190339232718839, 6.34147883218152869946511147953, 7.42603074747073835458560840276, 7.73699463387108337849874974979, 8.649781892141827627710348391573, 9.545771766541307161250340286341
