| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s + 164·7-s − 64·8-s + 81·9-s + 100·10-s + 720·11-s + 144·12-s + 698·13-s − 656·14-s − 225·15-s + 256·16-s − 2.22e3·17-s − 324·18-s + 356·19-s − 400·20-s + 1.47e3·21-s − 2.88e3·22-s − 1.80e3·23-s − 576·24-s + 625·25-s − 2.79e3·26-s + 729·27-s + 2.62e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.26·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.79·11-s + 0.288·12-s + 1.14·13-s − 0.894·14-s − 0.258·15-s + 1/4·16-s − 1.86·17-s − 0.235·18-s + 0.226·19-s − 0.223·20-s + 0.730·21-s − 1.26·22-s − 0.709·23-s − 0.204·24-s + 1/5·25-s − 0.809·26-s + 0.192·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.458841352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458841352\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 - 164 T + p^{5} T^{2} \) |
| 11 | \( 1 - 720 T + p^{5} T^{2} \) |
| 13 | \( 1 - 698 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2226 T + p^{5} T^{2} \) |
| 19 | \( 1 - 356 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1800 T + p^{5} T^{2} \) |
| 29 | \( 1 - 714 T + p^{5} T^{2} \) |
| 31 | \( 1 - 848 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11302 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9354 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5956 T + p^{5} T^{2} \) |
| 47 | \( 1 + 11160 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14106 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7920 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13450 T + p^{5} T^{2} \) |
| 67 | \( 1 + 65476 T + p^{5} T^{2} \) |
| 71 | \( 1 - 34560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 86258 T + p^{5} T^{2} \) |
| 79 | \( 1 + 108832 T + p^{5} T^{2} \) |
| 83 | \( 1 - 10668 T + p^{5} T^{2} \) |
| 89 | \( 1 - 10818 T + p^{5} T^{2} \) |
| 97 | \( 1 - 4418 T + p^{5} T^{2} \) |
| 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
−15.94555312174174865711132164202, −14.86302060930957981860331875808, −13.77877578714911157333514521648, −11.83168852078855017053567342381, −10.95744662170537519521873514322, −9.064107112878465724010180757793, −8.282969272695719530922799699845, −6.70048339117703975819519283748, −4.09924323660202214116565515612, −1.57672569632174089259606546347,
1.57672569632174089259606546347, 4.09924323660202214116565515612, 6.70048339117703975819519283748, 8.282969272695719530922799699845, 9.064107112878465724010180757793, 10.95744662170537519521873514322, 11.83168852078855017053567342381, 13.77877578714911157333514521648, 14.86302060930957981860331875808, 15.94555312174174865711132164202
