| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 4·11-s + 12-s + 13-s + 2·14-s + 16-s − 8·17-s − 18-s − 6·19-s − 2·21-s − 4·22-s − 6·23-s − 24-s − 26-s + 27-s − 2·28-s − 4·29-s − 32-s + 4·33-s + 8·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.37·19-s − 0.436·21-s − 0.852·22-s − 1.25·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.377·28-s − 0.742·29-s − 0.176·32-s + 0.696·33-s + 1.37·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.907841935382063111565042358126, −8.275106945738770193737766007203, −7.21926473255081566972637327843, −6.51341800580325524726586052920, −6.04011696824586895486685368245, −4.34999282857824787062750680049, −3.80974243367827903050974784654, −2.54833875126849207240384424642, −1.70830709219379145242715005824, 0,
1.70830709219379145242715005824, 2.54833875126849207240384424642, 3.80974243367827903050974784654, 4.34999282857824787062750680049, 6.04011696824586895486685368245, 6.51341800580325524726586052920, 7.21926473255081566972637327843, 8.275106945738770193737766007203, 8.907841935382063111565042358126
