| L(s) = 1 | − 2i·2-s + i·3-s − 2·4-s + 2·6-s − 5i·7-s − 9-s + 11-s − 2i·12-s − 2i·13-s − 10·14-s − 4·16-s − i·17-s + 2i·18-s + 19-s + 5·21-s − 2i·22-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s + 0.577i·3-s − 4-s + 0.816·6-s − 1.88i·7-s − 0.333·9-s + 0.301·11-s − 0.577i·12-s − 0.554i·13-s − 2.67·14-s − 16-s − 0.242i·17-s + 0.471i·18-s + 0.229·19-s + 1.09·21-s − 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213258106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213258106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| 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
−9.548766757501507698246424256944, −8.559541758158458095367011329787, −7.43488676905309528588313224795, −6.82464734474922463008327271495, −5.41081426384085138124804158955, −4.36002682787085093088138193913, −3.75895657612691884878267342316, −3.07564879122154319068832060598, −1.59333984936815195897336748818, −0.48458776575377232738719977766,
1.89387420170770766800380050930, 2.85061582980062512808896909837, 4.49112833241380922224768471874, 5.44590430097588137327209655755, 6.03442508510505748600502576477, 6.62360351443807592976756319500, 7.51918595249373066857123824590, 8.333437702861934967332757652589, 8.919360813747291069389790560515, 9.413536435927328250175960390265
