L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 2·7-s + 8-s + 9-s + 10-s + 12-s − 4·13-s + 2·14-s + 15-s + 16-s − 17-s + 18-s − 4·19-s + 20-s + 2·21-s + 24-s + 25-s − 4·26-s + 27-s + 2·28-s + 6·29-s + 30-s − 4·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.839872482\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.839872482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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
−10.91088498824282291972206366659, −10.11499143294052566213295385110, −9.103492689940007365990822328112, −8.129869086703898087756606920019, −7.24701119473791686481061072294, −6.23603513691922404166207627130, −5.03732857913234617258396270092, −4.30345448606923584518040820992, −2.85161342660248480872366985845, −1.85001277183392738194431289078,
1.85001277183392738194431289078, 2.85161342660248480872366985845, 4.30345448606923584518040820992, 5.03732857913234617258396270092, 6.23603513691922404166207627130, 7.24701119473791686481061072294, 8.129869086703898087756606920019, 9.103492689940007365990822328112, 10.11499143294052566213295385110, 10.91088498824282291972206366659