L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 6·11-s + 12-s + 5·13-s + 14-s − 15-s + 16-s − 18-s − 20-s − 21-s − 6·22-s + 6·23-s − 24-s + 25-s − 5·26-s + 27-s − 28-s + 6·29-s + 30-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 + 1.80·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.451702655\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.451702655\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−16.46845364747815, −16.00100234604119, −15.47835853311596, −14.90383992513259, −14.25427224088457, −13.84259539105867, −12.97788340424280, −12.54240307344790, −11.66926020405516, −11.35365438061613, −10.70986170433115, −9.944237018575117, −9.222605614943090, −8.982970336234498, −8.338788970520050, −7.732371084853932, −6.919078285997604, −6.421643957740092, −5.943241700041822, −4.570975010242444, −4.041945390825184, −3.272760778713735, −2.655063376838759, −1.323206996198386, −0.9488138736495322,
0.9488138736495322, 1.323206996198386, 2.655063376838759, 3.272760778713735, 4.041945390825184, 4.570975010242444, 5.943241700041822, 6.421643957740092, 6.919078285997604, 7.732371084853932, 8.338788970520050, 8.982970336234498, 9.222605614943090, 9.944237018575117, 10.70986170433115, 11.35365438061613, 11.66926020405516, 12.54240307344790, 12.97788340424280, 13.84259539105867, 14.25427224088457, 14.90383992513259, 15.47835853311596, 16.00100234604119, 16.46845364747815