L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 20-s − 21-s − 22-s + 4·23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−15.32021076479441, −14.89777095898585, −14.68809117401870, −13.65201813209794, −13.35732282933579, −12.80066806135887, −12.03695474236240, −11.68606405248856, −11.11448945029934, −10.36739761641927, −10.03628311598272, −9.291247327957131, −8.954466066385689, −8.374633558354629, −7.795558932696461, −7.279521522238329, −6.657151873419574, −6.248412372062437, −5.274049510919459, −4.597994351955811, −3.906958971425644, −3.171579459466806, −2.669456346054175, −1.820354172990721, −0.9884331412128639, 0,
0.9884331412128639, 1.820354172990721, 2.669456346054175, 3.171579459466806, 3.906958971425644, 4.597994351955811, 5.274049510919459, 6.248412372062437, 6.657151873419574, 7.279521522238329, 7.795558932696461, 8.374633558354629, 8.954466066385689, 9.291247327957131, 10.03628311598272, 10.36739761641927, 11.11448945029934, 11.68606405248856, 12.03695474236240, 12.80066806135887, 13.35732282933579, 13.65201813209794, 14.68809117401870, 14.89777095898585, 15.32021076479441